Post on 08-Jun-2022
1
Chapter 7 Quadrilaterals
7A p.2
7B p.16
7C p.30
7D p.37
Chapter 8 More about 3-D Figures
8A p.49
8B p.57
8C p.66
8D p.77
Chapter 9 Area and Volume (III)
9A p.89
9B p.104
9C p.120
9D p.131
9E p.137
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2
F3B: Chapter 7A
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3
Maths Corner Exercise
7A Level 1
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4
Book 3B Lesson Worksheet 7A (Refer to §7.2)
7.2A Properties of Parallelograms
(a) Definition of a parallelogram:
a quadrilateral with two pairs of parallel opposite sides
(b) Properties of parallelograms:
(i) The opposite sides of a parallelogram are equal.
i.e. In the figure, AB = DC and AD = BC.
[Reference: opp. sides of //gram]
(ii) The opposite angles of a parallelogram are equal.
i.e. In the figure, ∠A = ∠C and ∠B = ∠D.
[Reference: opp. ∠s of //gram]
(iii) The diagonals of a parallelogram bisect each other.
i.e. In the figure, AE = EC and BE = ED.
[Reference: diagonals of //gram]
1. Fill in the blanks for each of the following parallelograms.
(a) (b)
PTR and STQ are straight lines.
(c) (d)
WMY and XMZ are straight lines.
A B
C D
7
48°
Q R
S P
4 9 T
52°
7
E F
G H 10
W X
Y Z
M
9
5 6
A B
C D
A B
C D
E
A B
C D
The opposite side of DC is ____.
The opposite angle of ∠A is ∠__.
The diagonals PR and SQ bisect each
other.
i.e. PT = ____ and ST = ____.
5
○○○○→→→→ Ex 7A 1−3
Example 1 Instant Drill 1 In the figure, ABCD is a parallelogram. AC and
BD intersect at E. Find the unknowns.
Sol ∠DCB = ∠DAB (opp. ∠s of //gram)
x = 25° + 27°
= 52° DE = EB (diagonals of //gram)
2y = y + 6
y = 6
In the figure, RUST is a
parallelogram. RS and
TU intersect at O. Find
the unknowns.
Sol RT = ____ ( of //gram)
h = ___________
=
Find the unknowns in each of the following parallelograms. [Nos. 2−−−−5]
2.
PTR and QTS are straight lines.
3.
4.
BEC is a straight line.
5.
CEB is a straight line.
○○○○→→→→ Ex 7A 4−9
Q R
S P
2x
x + y 10 9 T
Q R
S P
y
2x − 3°
x + 34°
E A
B
C
D
p q
25° 21°
A
B
C
D
q 3p
2p E
60°
AD // BC
∵ PQ // SR ∴ ∠Q + ∠R
= ______
A
B C
D
2y
y + 6 x
27°
25°
E
R
U
S
T
k
h
h − 2
3h − 14
O
6
Example 2 Instant Drill 2
In the figure, PQRS is a parallelogram. QTR is
a straight line. Find the unknowns.
Sol ∠PTQ + 112° = 180° (adj. ∠s on st. line)
∠PTQ = 68°
In △PQT,
∵ PQ = PT (given)
∴ ∠PQT = ∠PTQ (base ∠s, isos. △)
x = 68°
∠S = ∠Q (opp. ∠s of //gram)
y = x
= 68°
In the figure, ABCD is a parallelogram. Find
the unknowns.
Sol ∠DAB = ∠______ ( )
=
In △ABD,
∵ BD = ____ ( )
∴
6. In the figure, CEFG is a parallelogram. CGH and DEF are
straight lines. Find x.
○○○○→→→→ Ex 7A 10−12
7. In the figure, QTR is a straight line. PSRT is a parallelogram.
If PS = 9 cm and QR = 15 cm, find
(a) the length of PT,
(b) the perimeter of quadrilateral PQRS.
○○○○→→→→ Ex 7A 13, 14
Q R
S P y
112° x
T
B C
D A
y
70° x
Is △FGH isosceles? C
E F
G H
x
D 75°
Q R
S P
T
△PQT is an
__________
triangle.
7
7.2B Tests for Parallelograms
Conditions for identifying a parallelogram:
(a) Both pairs of opposite sides of a quadrilateral are equal.
i.e. In the figure, if AB = DC and AD = BC,
then ABCD is a parallelogram.
[Reference: opp. sides equal]
(b) Both pairs of opposite angles of a quadrilateral are equal.
i.e. In the figure, if ∠A = ∠C and ∠B = ∠D,
then ABCD is a parallelogram.
[Reference: opp. ∠s equal]
(c) The diagonals of a quadrilateral bisect each other.
i.e. In the figure, if AE = EC and BE = ED,
then ABCD is a parallelogram.
[Reference: diags. bisect each other]
(d) One pair of opposite sides of a quadrilateral are equal
and parallel.
i.e. In the figure, if AB // DC and AB = DC,
then ABCD is a parallelogram.
[Reference: 2 sides equal and //]
8. Determine whether each of the following quadrilaterals must be a parallelogram. If yes, write
down the reasons.
(a) (b)
� Yes, ____________________ � Yes, ____________________
� No � No
(c) (d)
� Yes, ____________________ � Yes, ____________________
� No � No
(e) (f)
� Yes, ____________________ � Yes, ____________________
A
B C
D
135°
45°
45°
135°
E
F G
H 8
8
L
K J
I 10
10
6 6
U T
R S
12
12
8
8
K
L M
N
4 4
P
Q
R
S 11
11
A B
C D
A B
C D
A B
C D
A B
C D
E
8
� No � No
Example 3 Instant Drill 3 In the figure, AC and BD intersect at E. Prove
that ABCD is a parallelogram.
Refer to the figure. Prove that EFGH is a
parallelogram.
Sol AE = EC = 7
BE = ED = 6
∴ ABCD is a
parallelogram.
given
given
diags. bisect each
other
Sol EF = ____ = 3
EH = ____ = ____
∴ ____________
____________
given
given
___________________
___________________
9. Refer to the figure. Prove that PQRS is a
parallelogram.
10. Refer to the figure. Prove that ABCD is a
parallelogram.
11. In the figure, PST is a straight line. Prove that PQRS is
a parallelogram.
○○○○→→→→ Ex 7A 15
A
B C
D
E
6 6
7 7
E
F G
H 4 4
3 3
P
Q
R
S 80°
80°
100°
100°
A
B C
D 10 10 133° 47°
Is AB parallel
to DC?
P
Q
R
S
T
We can also identify a
parallelogram by
its definition (see
P.7A-1).
9
Example 4 Instant Drill 4
Refer to the figure.
(a) Find x.
(b) Prove that ABCD
is a parallelogram.
Refer to the figure.
(a) Find a.
(b) Prove that PQRS
is a parallelogram.
Sol (a) (x + 17°) + 113° +
(2x − 33°) + 113° = (4 − 2) × 180°
Sol (a) 104° + ( ) +
( ) + ( ) =
[( ) − ( )] ×
180° (∠ sum of polygon)
3x + 210° = 360°
3x = 150°
x = 50°
( )
=
(b) ∠B = ∠D = 113°
∠A = x + 17°
= 50° + 17°
= 67°
∠C = 2x − 33°
= 2 × 50° − 33°
= 67°
∴ ∠A = ∠C = 67°
∴ ABCD is a
parallelogram.
given
from (a)
from (a)
opp. ∠s equal
(b) ∠P = ∠__ = ____
∠Q = 3a − 29°
=
given
12. In the figure, AC and BD intersect at E.
(a) Find y.
(b) Prove that ABCD is a parallelogram.
○○○○→→→→ Ex 7A 18, 19
A
B
C
D
113°
x + 17°
2x − 33°
113°
P
Q R
S
104°
2a + 6°
3a − 29°
104°
A
B
C
D 21
20 y
29
20 E
10
Level Up Questions
13. In the figure, AEB and DFC are straight lines. AEFD is
a parallelogram. Prove that ABCD is a parallelogram.
14. Refer to the figure.
(a) Prove that △BDA ≅ △DBC.
(b) Hence, prove that ABCD is a parallelogram.
Note the difference
between the
‘properties’ and
‘conditions’ of
parallelograms.
A
B C
D
E F
A
B
C
D
11
A B
C D
5 cm
8 cm
y cm
x cm
S
R
Q
P
10 cm 8x cm
5x cm y cm
A B
C D
3y cm 9 12 cx A
B C
D
2y + 15° x
65°
E F
G H
70° 80°
x
S
P
R
Q
3x 2x
y 125°
E
D
C B
A (2x + 3) cm
(4x − 1) cm 5 c(x + 3y) cm
x 2x
72°
9 cm
O
P
Q
R
S
3
ycm
New Century Mathematics (2nd Edition) 3B
7 Quadrilaterals
Level 1
Find the unknown(s) in each of the following parallelograms. [Nos. 1−−−−8]
1. 2.
3. 4.
AEC and BED are straight lines.
5. 6.
PTQ is a straight line.
7. 8.
AEC and BED are straight lines.
POR and QOS are straight lines.
Consolidation Exercise
7A
12
9. In the figure, ABCD is a parallelogram. If CA = CD, find x.
10. In the figure, PQRS is a parallelogram. TPQ is a straight line. If
RQ = ST, find the perimeter of quadrilateral QRST.
11. In the figure, AGCB and CFED are parallelograms. BCD and
GFC are straight lines. Find x and y.
12. In the figure, △ABD ≅ △CDB. Prove that ABCD is a
parallelogram.
13. In the figure, PQRS and PQTU are parallelograms. Prove that
(a) SR = UT,
(b) RSUT is a parallelogram.
14. In the figure, ABCD is a parallelogram. AGC and BFGED are
straight lines. It is given that EG = FG. Prove that AFCE is a
parallelogram.
15. In the figure, PQRS is a parallelogram. PSU and TQR are
straight lines. It is given that SU = QT. Prove that PTRU is a
parallelogram.
16. Refer to the figure.
(a) Find x.
(b) Prove that ABCD is a parallelogram.
40°
3 x c7 6
x + 40°
1196° − x
6 cm
8 cm
9 cm
13
Level 2
17. In the figure, ABCD is a parallelogram. CEB is a straight line.
It is given that AC = AD and AE ⊥ CB.
(a) Find ∠ADC.
(b) Find ∠BAE.
18. In the figure, ABCD is a parallelogram. FBA, FED and CEB
are straight lines. It is given that FB = EB.
(a) Find ∠CDF.
(b) Find ∠FAD.
19. In the figure, ABCD and AEDF are parallelograms. AEC and BED
are straight lines. Find x and y.
20. In the figure, PQRS is a parallelogram. If PR = 8 cm and the
perimeter of △PRS is 30 cm, find the perimeter of PQRS.
21. In the figure, ABCD is a parallelogram. AC and BD
intersect at E. CF ⊥ FB and BD ⊥ DA. If BE = 4 cm,
AD = 15 cm and FC = 9 cm, find the perimeter of pentagon
ABFCD.
22. In the figure, RTQU is a parallelogram. PTR and STQ are
straight lines. It is given that RU = ST and QU = PT. Prove that
PQRS is a parallelogram.
23. In the figure, ABCD is a parallelogram. DFC and CEB are
straight lines. It is given that ∠CEG = ∠CBA and FC = GE.
(a) Prove that CFGE is a parallelogram.
(b) If AD = 12 cm and BE = 4 cm, find FG.
50 50y cm 10 cm 25 35 8 cm
14
24. Refer to the figure.
(a) Find the value of a.
(b) Prove that ABCD is a parallelogram.
(c) Find the value of b.
25. In the figure, ABCD is a parallelogram. AFGEC and BGD are
straight lines. It is given that ∠DFG = ∠BEG.
(a) Prove that △DGF ≅ △BGE.
(b) Prove that BEDF is a parallelogram.
26. In the figure, ABCD is a parallelogram. AED and BFC are
straight lines. What condition(s) about △ABF and △CDE
should be added so as to make AFCE a parallelogram? Explain
your answer.
a° + °
5a° + °
4a° − °
12 a + 9 3b − 1
15
Answer
Consolidation Exercise 7A
1. x = 8, y = 5
2. x = 2, y = 16
3. x = 9, y = 4
4. x = 65°, y = 50°
5. x = 30°
6. x = 25°, y = 50°
7. x = 2, y = 1
8. x = 24°, y = 27
9. 70°
10. 40 cm
11. x = 10, y = 120°
16. (a) 28°
17. (a) 65° (b) 25°
18. (a) 50° (b) 80°
19. x = 120°, y = 10
20. 44 cm
21. 70 cm
23. (b) 8 cm
24. (a) 20 (c) 10
26. △ABF ≅ △CDE
16
F3B: Chapter 7B
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7B Level 1
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18
Book 3B Lesson Worksheet 7B (Refer to §7.3)
7.3A Properties of Rhombuses
(a) Definition of a rhombus:
a quadrilateral with 4 equal sides
(b) Properties of rhombuses:
Property All properties of a
parallelogram.
Diagonals are
perpendicular to
each other.
Each interior angle is
bisected by a
diagonal.
Reference property of rhombus
Example 1 Instant Drill 1
In the figure, ABCD is a
rhombus. AEC and BED
are straight lines. Find the
unknowns.
In the figure, PQRS is a
rhombus. PTR and STQ
are straight lines. Find the
unknowns.
Sol ∵ AB = BC (by definition)
∴ x = 9
∵ AC ⊥ BD (property of rhombus)
∴ y = 90°
Sol m = ( of rhombus)
∵ RT = ____ ( of rhombus)
∴ n =
Find the unknowns in each of the following rhombuses. [Nos. 1−−−−2]
1.
PTR and STQ are straight lines.
2.
AEC and BED are straight lines.
○○○○→→→→ Ex 7B 1, 6, 12, 13
P
Q
R
S
T m
4
5
n A
B
C
D
x y 52°
E
A rhombus has 4 equal
sides.
A rhombus is a
kind of
parallelogram.
A
B
C
D
9
x
y
E
P
Q R
S
T
m
56°
n
4
Including:
(i) Opposite sides
are equal.
(ii) Opposite angles
are equal.
(iii) Diagonals bisect
each other.
19
7.3B Properties of Rectangles
(a) Definition of a rectangle:
a quadrilateral with 4 equal interior angles
(b) Properties of rectangles:
Property
All properties of
a parallelogram.
All the interior
angles are right
angles.
Diagonals are equal. Diagonals bisect
each other into four
equal line segments.
AC = BD
Reference property of rectangle
Example 2 Instant Drill 2
In the figure, ABCD is a
rectangle. AEC and BED
are straight lines. Find
the unknowns.
In the figure, PQRS is a
rectangle. PTR and STQ
are straight lines. Find
the unknowns.
Sol ∵ DE = EB (property of rectangle)
∴ x = 5
∵ AC = BD (property of rectangle)
∴ y = 5 + x
= 5 + 5
= 10
Sol ∠SRQ = _____ ( of rectangle)
( ) + m = ( )
m =
∵ ST = ____ ( of rectangle)
∴ n =
Find the unknowns in each of the following rectangles. [Nos. 3−−−−4]
3.
AEC and BED are straight lines.
4.
EKG and HKF are straight lines.
○○○○→→→→ Ex 7B 2, 7, 10
A
B C
D
A
B C
D
5
x y
z
6 E
E
G
H
m 136°
n
F
K
A
B C
D
E y
x
5
P
Q R
S
m 35°
n 7
T
Which kind of
triangle is △BCD?
Which kind of
triangle is △GHK?
20
7.3C Properties of Squares
(a) Definition of a square:
a quadrilateral with four equal sides and four equal interior angles
(b) Properties of squares:
Property
All properties of a
rectangle.
All properties of a
rhombus.
Angle between a side
and a diagonal is 45°.
Reference property of square
Example 3 Instant Drill 3
In the figure, ABCD is a
square. Find the unknowns.
In the figure, PQRS is a
square. PTR and STQ
are straight lines. Find
the unknowns.
Sol ∵ CD = BC (by definition)
∴ x = 10
y = 45° (property of square)
z = 90° (property of square)
Sol ∵ PS = ____ ( )
∴ 4p =
∵ SQ ⊥ ____ ( )
∴ n =
Find the unknowns in each of the following squares. [Nos. 5−−−−6]
5.
AED is a straight line.
6.
PTS, PWR, TWX and QXR are straight lines.
○○○○→→→→ Ex 7B 3, 8, 11
45°
A
B C
D
x
y
30°
E P
Q R
S
W
m
k
n
64°
T
X
A
B C
D
x
y
10
z
A square is a kind
of rectangle.
P
Q R
S
T n p + 9
4p
A square is a
kind of
∵ A square is a kind of
parallelogram. ∴ PS // ____
21
A D
C B
7.3D Properties of Other Special Quadrilaterals
I. Properties of Trapeziums
(a) Definition of a trapezium:
a quadrilateral with only one pair of parallel oppposite sides
(b) The figure shows an isosceles trapezium.
It has the following properties:
(i) ∠B = ∠C,
(ii) ∠A = ∠D,
(iii) AC = BD.
Find the unknowns in each of the following isosceles trapeziums. [Nos. 7−−−−8]
7.
∠P + ∠___ = 180° (_____ ∠s, PS // QR)
=
∵ PQRS is an isosceles trapezium. (given)
∴ ∠R = ∠__
k =
∠S = ∠__
=
8.
AEC and BED are straight lines.
∵ ABCD is an isosceles trapezium. (given)
∴
○○○○→→→→ Ex 7B 4(a)
Q R
S P
78°
m n
k 2 5
4
y
E
A
B
D
C
x
22
II. Properties of Kites
(a) Definition of a kite:
a quadrilateral with two pairs of equal adjacent sides
(b) The figure shows a kite, where AC and BD intersect
at O. It has the following properties:
(i) ∠ABC = ∠ADC, ∠BAC = ∠DAC,
∠BCA = ∠DCA;
(ii) BO = DO;
(iii) AC ⊥ BD.
[Reference: property of kite]
In each of the following, ABCD is a kite. AC and BD intersect at O. Find m and n. [Nos. 9−−−−10] 9.
OB = ____ (property of kite)
m =
∠DAC = ∠______ (property of ____)
n =
10.
In each of the following, PQRS is a kite, where PQ = PS and RQ = RS. Find x and y. [Nos. 11−−−−12]
11. 12.
○○○○→→→→ Ex 7B 5
A
B
C
D O
m
21°
n
4
5
A
B
C
D m 41° n
103° O
P
Q
R
S
54°
x
110°
y
P
Q
R
S
x 48°
y
95°
A
B
C
D O
Recall:
Sum of the interior
angles of a polygon
= (n − 2) × _____
(∠ sum of polygon)
� In fact,
△ABC ≅ △ADC
(SSS).
23
���� ‘Explain Your Answer’ Question
13. In the figure, ABCD is a parallelogram. BCE is a straight line.
(a) Find ∠BCD.
(b) By definition, determine whether ABCD is a rectangle.
Explain your answer.
Level Up Question
14. Refer to the figure. Find DC.
(Hint: Draw a perpendicular line from D to BC.)
Trapezium ABCD has a
side perpendicular to
its two bases. It is
called a right-angled
trapezium.
A
B C
D
E
A
B C
D
12 cm
15 cm
50 cm
24
12 cm
x cm 3y cm
(z − 3) cm
5010 38
65 3y + 1
16 − 255 9 cm (z − 1) cm
New Century Mathematics (2nd Edition) 3B
7 Quadrilaterals
Level 1
1. Find the unknowns in each of the following rhombuses.
(a) (b)
(c) (d)
POR and SOQ are straight lines. EOG and HOF are straight lines.
2. Find the unknowns in each of the following rectangles.
(a) (b)
(c) (d)
POR and SOQ are straight lines.
POR and SOQ are straight lines.
Consolidation Exercise
7B
25
10 cm (b + 2) cm
a cm
22 3r c9 cm
6556 2x
29 11 42
5 x y cm 12
3. Find the unknowns in each of the following squares.
(a) (b)
(c) (d)
4. Find the unknowns in each of the following trapeziums.
(a) (b)
PTQ is a straight line.
5. Find the unknowns in each of the following kites.
(a) (b)
AEC and DEB are straight lines.
26
6. In the figure, ABCD is a rhombus. AC and BD intersect at E.
It is given that AC = 16 cm and BD = 30 cm. Find the
perimeter of rhombus ABCD.
7. In the figure, ABCD is a rectangle. AC and BD intersect at E.
It is given that AD = 16 and BE = 10. Find AB.
8. In the figure, PQRS is a square. PR and QS intersect at O. The
perimeter of PQRS is 36 cm.
(a) Find PQ.
(b) Find PO, correct to 3 significant figures.
9. In the figure, ABCD is a rectangle. E is a point on DB such that
EC = BC.
(a) Find ∠CEB.
(b) Find ∠ECD.
10. In the figure, ABCD is a square. DFB and AEF are straight lines. It
is given that AB = EB and ∠EAB = 75°.
(a) Find ∠AFB.
(b) Find ∠DBE.
11. In the figure, PQRS is a rhombus. STQ is a straight line. U is a
point on RQ such that TU ⊥ RQ. It is given that ∠SPQ = 110°.
(a) Find ∠PQS.
(b) Find ∠UTS.
16 cm 30 cm
1070
7511
27
12. The figure shows a kite ABCD, where AB = AD and CB = CD. It is given
that ∠ADC = 140° and ∠CAD = 20°.
(a) Find ∠ACD.
(b) By definition, determine whether ABCD is a rhombus. Explain your
answer.
13. In the figure, PQRS is a rectangle. It is given that PS = 6 cm and the
perimeter of PQRS is 24 cm.
(a) Find SR.
(b) By definition, determine whether PQRS is a square. Explain
your answer.
Level 2
14. In the figure, ABCD is a rhombus. The area of ABCD is 96 cm2 and
DB = 12 cm. Find the perimeter of ABCD.
15. In the figure, PQRS is a rhombus. T is a point on SR such that TP
is the angle bisector of ∠SPR. It is given that ∠PTS = 63°.
(a) Find ∠TPR.
(b) Find ∠PQR.
16. The figure shows a trapezium ABCD, where AB // DC and AD
= BC. DCE is a straight line. It is given that
∠DAC = 32° and ∠BCE = 66°. Find ∠ACD.
17. In the figure, ABCD and BGFD are rectangles. AEC, BED and
FCG are straight lines. It is given that ∠BAC = 24°.
(a) Find ∠BEC.
(b) Find ∠FDC.
18. In the figure, ABCD is a rectangle and CEDF is a rhombus.
AFC and BFD are straight lines. It is given that AD = 18 cm and
DE = 15 cm.
(a) Find DC.
(b) Find the perimeter of pentagon ABCED.
14 206 c12 cm
63
2418 cm 15 cm
32 66
28
19. In the figure, ABC, AFD and EFB are straight lines. ABDE is a
parallelogram and BCDE is a square. It is given that ED = 3 cm.
(a) Find ∠EAB.
(b) Find the perimeter of quadrilateral ACDE.
(c) Find FD.
(Give your answers correct to 3 significant figures if necessary.)
20. In the figure, ABDE is a square and BCD is an equilateral triangle.
EFC is a straight line.
(a) Find ∠DCE.
(b) Find ∠BFE.
21. In the figure, ABCD is a square. BFE and DFC are straight
lines. It is given that ∠FBC = 16°, ∠FDE = 77° and DE =
12 cm.
(a) Prove that △BDE is isosceles.
(b) Find the area of square ABCD.
22. In the figure, ABCD is a parallelogram and ADEF is a
rectangle.
(a) Is it true that ADEF must be a square? Explain your
answer.
(b) If BC = AF, find ∠AEF.
16 77 12 cm
3 cm
29
Answer
Consolidation Exercise 7B
1. (a) x = 12, y = 4, z = 15
(b) x = 50°, y = 50°, z = 80°
(c) x = 6, y = 8, z = 10
(d) x = 90°, y = 52°, z = 52°
2. (a) x = 90°, y = 25°
(b) x = 2, y = 3
(c) x = 55°, y = 110°, z = 35°
(d) x = 3, y = 12, z = 10
3. (a) a = 10, b = 8
(b) x = 90°, y = 90°, z = 45°
(c) p = 23°, q = 112°, r = 3
(d) p = 25°, q = 45°, r = 70°
4. (a) x = 62°, y = 124°
(b) x = 5, y = 7
5. (a) x = 9, y = 90°, z = 29°
(b) x = 21°, y = 28°, z = 26°
6. 68 cm
7. 12
8. (a) 9 cm (b) 6.36 cm
9. (a) 70° (b) 50°
10. (a) 60° (b) 15°
11. (a) 35° (b) 125°
12. (a) 20° (b) yes
13. (a) 6 cm (b) yes
14. 40 cm
15. (a) 21° (b) 96°
16. 34°
17. (a) 48° (b) 66°
18. (a) 24 cm (b) 90 cm
19. (a) 45° (b) 16.2 cm (c) 3.35 cm
20. (a) 15° (b) 105°
21. (b) 72 cm2
22. (a) no (b) 45°
30
F3B: Chapter 7C
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 16
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 17
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 18
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
7C Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
7C Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
7C Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
7C Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
31
Book 3B Lesson Worksheet 7C (Refer to §7.4)
7.4 Proofs Related to Parallelograms
Example 1 Instant Drill 1 In the figure, ABCD is a parallelogram. Prove
that △ADE ≅ △CBF.
In the figure, ABCD is a parallelogram. Prove
that △AED ≅ △CFB.
Sol In △ADE and △CBF,
∠ADE = ∠CBF
AD = CB
∠DAE = ∠BCF
∴ △ADE ≅ △CBF
given
opp. sides of //gram
opp. ∠s of //gram
ASA
Sol In △AED and △CFB,
AE = ____
∠DAE = ∠____
given
_________ of //gram
1. In the figure. ABCD is a parallelogram.
AC, BD and EG intersect at F.
(a) Prove that △AEF ≅ △CGF.
(b) Prove that AE = CG.
2. In the figure. ABCD is a parallelogram.
AEFC is a straight line.
(a) Prove that △ADF ≅ △CBE.
(b) Prove that BE // FD.
A B
C D
E
F A
B C
D
E F
A
B C
D
E
F G
A
B C
D
E
F
32
3. In the figure, ABCF is a parallelogram.
Prove that ED // BC.
4. In the figure, ABCD is a rhombus and
AEFC is a straight line. Prove that
△ABE ≅ △CDF.
○○○○→→→→ Ex 7C 1–6
5. In the figure, ABCD is a rectangle.
(a) Prove that △ABE ≅ △DCE.
(b) Prove that △BEC is an isosceles
triangle.
6. In the figure, ABCD is a square.
(a) Prove that △ADE ≅ △CDF.
(b) Prove that ∠ADF = ∠CDE.
○○○○→→→→ Ex 7C 7–10
A
B C
D E
F
A
B
C
E
D
F
A
B
E D
C
A
B
E
D
F C
33
Level Up Questions
7. In the figure, ABCD is a parallelogram with a diagonal BEFD.
(a) Prove that △DAF ≅ △BCE.
(b) If BF = 5 cm and EF = 1 cm, find BD.
8. In the figure, ABCD is a square.
(a) Prove that △EBF ≅ △ECF.
(b) Prove that △EAD is an isosceles triangle.
A
B C
D
E
F
A
B C
E
D
F
34
New Century Mathematics (2nd Edition) 3B
7 Quadrilaterals
Level 1
1. In the figure, ADE, GFE, DFC and AGB are straight lines. ABCD
is a parallelogram and ∠DFE = ∠BCD. Prove that ED = EF.
2. In the figure, ABCD is a parallelogram. AEB and DFC are
straight lines. It is given that ∠ADE = ∠CBF. Prove that
ED // BF.
3. In the figure, AGD and EDC are straight lines. ABCD and
DEFG are both parallelograms. Prove that ∠DAB = ∠EFG.
4. In the figure, ABC is a straight line. ABDE is a rectangle and
BCDE is a parallelogram. Prove that AB = BC.
5. In the figure, ABCD is a parallelogram. M and N are the
mid-points of AD and BC respectively. Prove that
△ABN ≅ △CDM.
Consolidation Exercise
7C
35
6. In the figure, ABCD is a rhombus. E is a point lying on AC. Prove
that △ABE ≅ △ADE.
7. In the figure, ABC and FED are straight lines. ABEF and BCDE
are both squares.
(a) Prove that △BED ≅ △BEF.
(b) Prove that △FBD is a right-angled isosceles triangle.
8. In the figure, BPCS is a straight line. ABCD and PQRS are two
identical rectangles.
(a) Prove that △BPQ ≅ △SCD.
(b) Prove that BQ // DS.
Level 2
9. In the figure, ABCD is a rectangle. E is the mid-point of AD. AC
and BE intersect at F.
(a) Prove that △AFE ~ △CFB.
(b) Prove that FC = 2AF.
10. In the figure, ABCD and BFDE are both parallelograms. AC
intersects DE and FB at G and H respectively.
(a) Prove that △AGE ≅ △CHF.
(b) Prove that △AGD ≅ △CHB.
11. In the figure, ABCD is a square. AC and BD intersect at F. It is given
that △AED ≅ △AFD. Prove that AFDE is a square.
36
12. In the figure, ABCD is a rectangle. E and F are points on AB and DC
respectively. AC and EF intersect at G. It is given that AE = FC and
EB2 + BC2 = AE2. Prove that EF ⊥ AC.
13. In the figure, ABCD is a square. E and F are points on AB and BC
respectively. CE and DF intersect at G. It is given that AE = BF.
(a) Prove that △EBC ≅ △FCD.
(b) Prove that EC ⊥ FD.
14. In the figure, ABCD is a rhombus. QBDP is a straight line
and ∠QAB = ∠PCD.
(a) (i) Prove that AQ = CP.
(ii) Prove that AQ // CP.
(b) Prove that AQCP is a rhombus.
37
F3B: Chapter 7D
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 19
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 20
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 21
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 22
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 23
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 24
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
7D Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
7D Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
38
Maths Corner Exercise
7D Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
7D Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
39
Book 3B Lesson Worksheet 7D (Refer to §7.5)
7.5A Mid-point Theorem
In △ABC,
if AM = MB and AN = NC, then
(a) MN // BC,
(b) MN =
2
1BC.
[Reference: mid-pt. theorem]
Example 1 Instant Drill 1 In the figure, M and N are
the mid-points of AB and
AC respectively. Find the
unknowns.
In the figure, M and N are
the mid-points of BC and
AB respectively. Find the
unknowns.
Sol ∵ AM = MB
and AN = NC. (given)
∴ MN // BC (mid-pt. theorem)
x = ∠ACB (corr. ∠s, MN // BC)
= 44°
MN =
2
1BC (mid-pt. theorem)
y =
2
1× 10
= 5
Sol ∵ AN = ____
and BM = ____. (given)
∴ ____ // ____ (__________ theorem)
∴ x = ∠BMN (_________________)
=
MN =
2
1×____ (mid-pt. theorem)
=
Find the unknown(s) in each of the following figures. [Nos. 1−−−−2]
1.
ANC and AMB are straight lines.
2.
APB, AQC and CRD are straight lines.
○○○○→→→→ Ex 7D 1–6
A
B
N 60°
x 4 cm
C
M
4 cm
3 cm
3 cm
74°
A
B
P
C
Q
D
R
x cm
y cm
11 cm
4.5 cm
A
B
M N
C
A
B
M N
C 44°
x
y cm
10 cm B C
N
85° x
y cm
3 cm
A
M
Consider △ABC
and △ACD one
by one.
40
7.5B Intercept Theorem
In the figure,
if AB // CD // EF and AC = CE,
then BD = DF.
[Reference: intercept theorem]
Example 2 Instant Drill 2
In the figure, ACE and
BDF are straight lines.
Find x.
In the figure, PRT and
QSU are straight lines.
Find y.
Sol ∵ AB // CD // EF
and BD = DF = 8 cm. (given)
∴ AC = CE (intercept theorem)
x = 7
Sol ∵ PQ // ____ // ____
and QS = ____ = ( ) cm. (given)
∴ RT = ____ ( )
y =
Find the unknown(s) in each of the following figures. [Nos. 3−−−−4]
3.
PRT and QSU are straight lines.
4.
ABCD and EFGH are straight lines.
○○○○→→→→ Ex 7D 7–9
P Q
T
R S
U
5 cm
y cm
A
B
C
E
D H
G
F
2 cm
x cm
4 cm
3 cm
3 cm
y cm
A B
E
C D
F
L1 L2
A B
E
C D
F
8 cm
8 cm 7 cm
x cm P Q
T
R S 11 cm 11 cm
9 cm y cm U
41
In △ABC,
if AM = MB and MN // BC,
then AN = NC.
[Reference: intercept theorem]
Example 3 Instant Drill 3
In the figure, ABC and AED are straight lines.
Find ED.
In the figure, ABC and CDE are straight lines.
Find BC.
Sol ∵ AB = BC = 4 cm
and BE // CD. (given)
∴ ED = AE (intercept theorem)
= 2.5 cm
Sol ∵ CD = ____ = ( ) cm
and BD // ____. ( )
∴ BC = ____ ( )
=
Find the unknown(s) in each of the following figures. [Nos. 5−−−−6]
5.
ADB and AEC are straight lines.
6.
ABCD and AEFG are straight lines.
A
B
C E
D
10 cm
x cm
A
B E
C
D
F
G
7 cm 6 cm
6 cm
6 cm
x cm
y cm
A
B
M N
C
A
B
C
E
D
4 cm
4 cm
2.5 cm 4.5 cm
7 cm
7 cm
A
B
C
E
D
42
7.
PQR, PTS and RTU are straight lines.
8.
ADB and CEB are straight lines.
○○○○→→→→ Ex 7D 10–13
9. In the figure, ABC and AED are straight lines.
(a) Find AE. (b) Find CD.
○○○○→→→→ Ex 7D 14
���� ‘Explain Your Answer’ Question
10. In the figure, X, Y and Z are the mid-points of PQ, PR and PS
respectively. A student claims that if XY = YZ, then QR = RS. Do
you agree? Explain your answer.
6 cm 10 cm
6 cm
x cm
y cm
P
Q
R S
T
U
5 cm
A
B
E
5 cm
C
D x cm
A
B
C
E
D
9 cm 12 cm
P
Q R
S X
Y
Z
Consider △PRS
and △PRU one
Do not mix up the mid-point theorem with
the intercept theorem.
Mid-point theorem: Intercept theorem:
(a) BE // CD AE = ED
(b) BE =2
1CD
Is AC parallel
to DE?
A
C D
E B
A
C D
B E
43
Level Up Questions
11. In the figure, ABC, AED and CEF are straight lines.
(a) Prove that AB = BC.
(b) Find BE.
12. In the figure, ABCD and AGFE are straight lines.
(a) Find CF.
(b) Find the perimeter of quadrilateral BCFG.
A
B
C
E
D
F
30 cm
Is AF parallel to
BE?
7 cm A
B
C
E D
F
G 7 cm
14 cm 18 cm
10 cm
44
5 cm
5 cm
8 cm 8 cm
9 cm x cm
12 cm y c
65 6572x cm 15 cm
13 85
New Century Mathematics (2nd Edition) 3B
7 Quadrilaterals
Level 1
Find the unknown(s) in each of the following figures. [Nos. 1−−−−5]
1. 2.
APB and AQC are straight lines. AMB and ANC are straight lines.
3. 4.
CDA and CEB are straight lines. ADB and AEC are straight lines. 5.
APB and AQC are straight lines.
Consolidation Exercise
7D
45
x c4 c x c2 c2 c2 c 3 c y c
x cm 5 cm 2x cm
6 cm 6 cm y cm
x cm y cm
9 cm 8 cm
11 cm 11 cm
y cm 5 cm
Find the unknown(s) in each of the following figures. [Nos. 6−−−−10]
6. 7.
ACE and BDF are straight lines. ABCD and EFGH are straight lines.
8. 9.
ABCD and AGFE are straight lines. ACE, AKF and BDF are straight lines.
10.
ACF, BDF and BEG are straight lines.
11. In the figure, ACE, ADF and BDE are straight lines. It is given
that AB // CD // EF and AD = DF. Find x and y.
6 cm x cm
46
12. In the figure, AEB, AFC, AGD, EFG and BCD are straight
lines. It is given that AF = FC.
(a) Prove that EG // BD.
(b) Find GD and FG.
Level 2
13. In the figure, AGC, DGE and BECF are straight lines. It
is given that BA // ED and CA // FD.
(a) Find GE.
(b) Prove that BE = EC = CF.
14. In the figure, G is the mid-point of AC, while F is the
mid-point of both BC and AE. ABD is a straight line.
(a) Prove that BC // DE.
(b) Find ∠EAD.
15. In the figure, ACE and BDF are straight lines. It is given that
AB // CD // EF and AC = CE. Find DF and CD.
(Hint: Join AF or BE.)
16. In the figure, E is the mid-point of AB and AD // EF // BC. AC
and BD intersect at F.
(a) Prove that AF = FC and BF = FD.
(b) Prove that ABCD is a parallelogram.
5 cm 5 cm 3 cm
8 cm
3 cm 4 cm 4 cm
6 cm
62 824 c8 c
22 cm
16 cm
6 c
47
17. In the figure, D, E and F are the mid-points of AB, BC and AC
respectively.
(a) Prove that △BED ≅ △FDE.
(b) Prove that ∠DBE = ∠EFD.
18. In the figure, D and E are the mid-points of AB and AC
respectively. DC and BE intersect at F.
(a) Prove that △FDE ~ △FCB.
(b) Is F the mid-point of BE? Explain your answer.
19. In the figure, AF and BE intersect CD at H and G
respectively. ACE and BDF are straight lines. It is given that
AB // CD // EF, AC = CE, EF = 26 cm and GH = 8 cm.
(a) Find AB.
(b) Find CD.
20. In the figure, ADBG, AEC, EFG and BFC are straight lines. It is
given that DE // BC and AD = DB = BG. Prove that BF =
3
1FC.
21. In the figure, ADB, AEGC, BCF and DGF are straight lines. It is
given that AD = DB, DE // BF and BC : CF = 2 : 1.
(a) Prove that △FCG ≅ △DEG.
(b) Find AG : GC.
8 c 26 cm
48
Answer
Consolidation Exercise 7D
1. 6
2. 18
3. 65°
4. 43°
5. x = 7.5, y = 48°
6. 4
7. x = 3, y = 3
8. x = 5, y = 12
9. x = 8, y = 9
10. x = 4.5, y = 5
11. x = 6, y = 8
12. (b) GD = 8 cm, FG = 1.5 cm
13. (a) 3 cm
14. (b) 36°
15. DF = 6 cm, CD = 19 cm
18. (b) no
19. (a) 10 cm (b) 18 cm
21. (b) 3 : 1
49
F3B: Chapter 8A
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 1
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 2
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
8A Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8A Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8A Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8A Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
50
Book 3B Lesson Worksheet 8A (Refer to §8.1)
[In this worksheet, the contents of Questions 1, 5–7 are beyond the scope of the curriculum.]
8.1A Reflectional Symmetry of 3-D Figures
If a solid can be divided by a plane into two identical parts and these two parts are
the mirror images of each other, the solid is said to have reflectional symmetry.
The plane is called the plane of reflection.
e.g.
1. In each of the following, determine whether the shaded plane is a plane of reflection of the
given 3-D figure.
(a)
(yes / no)
(b)
(yes / no)
(c)
(yes / no)
(d)
(yes / no)
(e)
(yes / no)
(f)
(yes / no)
� A solid may
have more than
one plane of
reflection.
51
Example 1 Instant Drill 1
Determine whether each of the following is a
plane of reflection of the cube ABCDEFGH.
(a) the plane containing EF and HG
(b) the plane passing through the mid-points
of AB, DC, EH and FG
Sol
(a) It is not a plane of reflection of the
cube.
(b) It is a plane of reflection of the cube.
Determine whether each of the following is a
plane of reflection of the regular tetrahedron
ABCD.
(a) the plane containing AB and passing
through the mid-point of CD
(b) the plane passing through the mid-points
of AB, BC and CD
Sol
(a) It (is / is not) a plane of reflection of
the regular tetrahedron.
(b) It (is / is not) a plane of reflection of
the regular tetrahedron.
2. Refer to the cube in Example 1. Determine whether each of the following is a plane of
reflection of the cube.
(a) the plane containing AF and BG (yes / no)
(b) the plane containing AB and EH (yes / no)
(c) the plane passing through the mid-points of AD, BC, FE and GH (yes / no)
3. Refer to the regular tetrahedron in Instant Drill 1. Determine whether each of the following is
a plane of reflection of the regular tetrahedron.
(a) the plane containing BC and passing through the mid-point of AD (yes / no)
(b) the plane passing through A and the mid-points of BC and CD (yes / no)
○○○○→→→→ Ex 8A 7, 8
A D
E F
G
B C
H
(a) (b)
D
E F
G
B C
H
A
E F
G H
D
B C
A
A
D B
C
(a) (b) A
D B
C
A
D B
C
� A cube has 6
identical square
faces.
� A regular
tetrahedron has
4 identical faces
of equilateral
triangles.
Indicate the plane
on the cube
first.
52
4. Draw all the planes of reflection of a cube in the following.
(Two of the planes of reflection have been drawn for you as examples.)
How many planes of reflection does a cube have?
5. Does each of the following 3-D figures have reflectional symmetry? If yes, write down the
number of planes of reflection. (a) (b) (c)
○○○○→→→→ Ex 8A 1–3
8.1B Rotational Symmetry of 3-D Figures
If a solid coincides with its original figure n times (n > 1) in one complete
revolution (i.e. 360°) about a straight line, the solid is said to have rotational
symmetry of order n. The line is called the axis of rotation.
e.g. Consider the right square prism below.
Note: An object can have different axes of rotation,
and the corresponding orders of rotational
symmetry can also be different.
axis of rotation
(n = 2)
P 1ℓ Consider the top face.
rotate
1
8
rotate
18
0°
1st coincidence 2nd coincidence
P
P
P
axis of rotation
(n = 4)
2ℓ
53
6. In each of the following, determine whether the straight line ℓ is an axis of rotation of the given
3-D figure. If yes, write down the corresponding order of rotational symmetry about ℓ .
(a)
(b)
� yes, order of rotational symmetry =
� no
� yes, order of rotational symmetry =
� no
(c)
(d)
� yes, order of rotational symmetry =
� no
� yes, order of rotational symmetry =
� no
○○○○→→→→ Ex 8A 4–6
Level Up Questions
7. The given figure shows a right triangular prism.
(a) Does it have reflectional symmetry?
(b) Does it have rotational symmetry?
8. In each of the following 3-D figures, X and Y are the mid-points of two edges, and Z is the
centre of the shaded face. Determine whether lines XY and AZ are the axes of rotation of the
given 3-D figures. If yes, write down the corresponding order of rotational symmetry.
(a) Cube ABCDEFGH (b) Regular tetrahedron ABCD
ℓ ℓ
ℓ
ℓ
A B
G F
E
D C
H
X
Z
Y
D
A B
C
Z
X
Y
54
P
Q
R
S
N
New Century Mathematics (2nd Edition) 3B
8 More about 3-D Figures
Level 1
In each of the following prisms, write down the number of planes of reflection. [Nos. 1−−−−2]
1. 2.
In each of the following 3-D figures, ℓ is the axis of rotation. Find the orders of rotational symmetry
of the figures about ℓ . [Nos. 3−−−−4]
3. 4.
5. In each of the following, determine whether the shaded plane is a plane of reflection of the cube
ABCDEFGH.
(a) (b) (c)
6. The figure shows a regular tetrahedron PQRS. N is the mid-point
of QR. Determine whether each of the following is a plane of
reflection of the regular tetrahedron.
(a) plane PQS
(b) plane PSN
ℓ
ℓ
A B
C D
E
F G
H
A
C D
E
F G
H
B A B
D
E
F G
H
C
Consolidation Exercise
8A
55
A B
C D
E
F G
H
P
R
Q
P
Q
R
S
U
Y Z
X
Level 2
7. The figure shows a cube ABCDEFGH. P, Q and R are the
mid-points of the edges AB, BC and EH respectively.
Determine whether each of the following is an axis of rotation
of the cube. If yes, write down its order of rotational symmetry.
(a) line PR
(b) line QR
(c) line AH
8. The figure shows a regular tetrahedron PQRS. X, Y and Z are
the mid-points of the edges PS, SR and QR respectively. U is
the centre of the face PQR.
Determine whether each of the following is an axis of rotation
of the tetrahedron. If yes, write down its order of rotational
symmetry.
(a) line SU
(b) line XY
(c) line XZ
(d) line UY
9. The figure shows a cube with one face in grey and five faces in
white. If we take the colours into account,
(a) how many plane(s) of reflection does the cube have?
(b) how many axis/axes of rotation does the cube have? Find
the corresponding order of rotational symmetry.
56
Answer
Consolidation Exercise 8A
1. 1
2. 2
3. 2
4. 3
5. (a) no (b) yes (c) no
6. (a) no (b) yes
7. (a) yes, 2 (b) no (c) yes, 3
8. (a) yes, 3 (b) no
(c) yes, 2 (d) no
9. (a) 4
(b) number of axis = 1, order = 4
57
F3B: Chapter 8B
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 3
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 4
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 5
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
8B Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8B Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8B Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8B Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
58
Book 3B Lesson Worksheet 8B (Refer to §8.2)
8.2 Nets of 3-D Figures
(a) A 3-D figure can be folded from different nets.
(b) To determine whether a given figure can be folded into a specified 3-D figure, we
have to consider the relative positions of the points, lines and faces.
1. Which of the following figures can be folded into the polyhedron on the right?
○○○○→→→→ Ex 8B 1, 2
Example 1 Instant Drill 1
The following shows a prism and its net.
Sketch another net of the prism.
Sol Another net is shown below:
(or other reasonable answers)
The following shows a tetrahedron and its net.
Sketch another net of the tetrahedron.
Sol
2. A cube is shown on the right. Sketch two different nets of the cube.
○○○○→→→→ Ex 8B 5, 6
A B C
59
Example 2 Instant Drill 2
If we fold the net above into a polyhedron,
(a) which point will coincide with point A?
(b) which line segment will be stuck together
with BC?
Sol
(a) Point G will coincide with point A.
(b) Line segment DC will be stuck
together with BC.
If we fold the net above into a polyhedron,
(a) which point will coincide with point A?
(b) which line segment will be stuck together
with FE?
Sol
(a) Point will coincide with point A.
(b) Line segment will be stuck
together with FE.
3. If we fold the net on the right into a polyhedron,
(a) which point(s) will coincide with point F ?
(b) which line segment will be stuck together with DE?
4. If we fold the net on the right into a polyhedron,
(a) which point(s) will coincide with point C?
(b) which line segment will be stuck together
with HI?
○○○○→→→→ Ex 8B 7, 8
H
B
D
A
C
F
G
E
A
G
C
D
B
H
E
F
A, G
H
B, D, F E
C
A
C
J B
E
D
I
F G
H
?
A B
C
D
E
J
I
H
G F
A
C
E B
K
N
D
J M
L
G
H
F
I
B
A
C
F
D
E
G
H
I
J
60
5. The net on the right is folded into a dice.
(a) Which number is on the face opposite to the face with the
number ‘3’?
(b) Which numbers are on the four faces adjacent to the face with
the number ‘1’?
○○○○→→→→ Ex 8B 9, 10
Level Up Questions
6. Sketch a net of the polyhedron on the right.
7. The net on the right is folded into a dice.
(a) Sketch the symbol on the face opposite to the face with ‘ ’.
(b) Sketch the symbols on the faces that will meet at ‘ ’.
1
2
3
4 5
6
61
New Century Mathematics (2nd Edition) 3B
8 More about 3-D Figures
Level 1
1.
Which of the following figures can be folded into the cuboid above?
2.
Which of the following figures can be folded into the polyhedron above?
3. Which of the following rectangular blocks can be folded from the
net shown on the right?
P Q R
P Q R
P Q R
Consolidation Exercise
8B
62
4. The following figure shows a 3-D figure with a triangular base and its net. Sketch another net of
the 3-D figure.
5. A square prism and its net are shown below. Sketch another net of the prism.
6. If we fold the net below into a polyhedron, which three points will coincide with one another?
7.
If we fold the net above into a polyhedron, which line segment will be stuck together with
(a) GH?
(b) DE?
B A
C
D
E
F
B
A
C
E F
D
H G
I
J
63
AAAA
BBBB CCCC DDDD
EEEE FFFF
37.5 cm 20 cm 15 cm
42.5 cm 30 cm Fig. A Fig. B
8.
The net above is folded into a dice. Which letter is on the face opposite to the face with the
letter ‘C’?
Level 2
9. Kelvin wants to make a paper file holder with an opening as shown in Fig. A. Label the
appropriate lengths for Kelvin’s net in Fig. B.
10. Sketch a net of each of the following polyhedra.
(a) (b)
11. Sketch a net of each of the following polyhedra.
(a) (b)
64
12. The following net is folded into a dice.
(a) What is the pattern on the face opposite to the face with ‘ ’?
(b) What are the patterns on the four faces adjacent to the face with ‘ ’?
13. The net below is folded into a dice.
Find the numbers on the faces that will meet at
(a) the point X,
(b) the point Y.
14. The net below is folded into a polyhedron.
(a) Which point(s) will coincide with C?
(b) Which line segment will be stuck together with GF?
1111
2222
3333
4444
5555
6666
C
A
E D
B H G
F
65
Answer
Consolidation Exercise 8B
1. P
2. Q
3. Q
6. B, D, F
7. (a) IH (b) BA
8. F
12. (a) ‘ ’
(b) ‘ ’, ‘ ’, ‘ ’, ‘ ’
13. (a) 3, 4, 5 (b) 1, 2, 3
14. (a) A, G (b) CD
66
F3B: Chapter 8C
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 6
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 7
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 8
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 9
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
8C Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8C Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8C Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8C Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
67
Book 3B Lesson Worksheet 8C (Refer to §8.3)
8.3A Front, Top and Side Views of 3-D Objects
We can observe a 3-D object from its front, its top or its side.
e.g.
Front view: Top view: Side view:
Example 1 Instant Drill 1
The 3-D object below is made up of four
identical cubes. Draw the front, top and side
views of the object.
Sol Front view: Top view:
Side view:
The 3-D object below is made up of four
identical cubes. Draw the front, top and side
views of the object.
Sol Front view: Top view:
Side view:
front
top
side
front
top
side front
top
side
68
The following 3-D objects are made up of five identical cubes. Draw the front, top and side views
of each of the objects. [Nos. 1–2] 1. Front view: Top view: Side view:
2. Front view: Top view: Side view:
○○○○→→→→ Ex 8C 1, 3–5
3. The 3-D object below is made up of four identical cubes and a triangular prism. Draw the front,
top and side views of the object.
Front view: Top view: Side view:
○○○○→→→→ Ex 8C 2, 6, 7
4. The 3-D object below is made up of six identical cubes and a half-cylinder. Draw the front, top
and side views of the object.
Front view: Top view: Side view:
○○○○→→→→ Ex 8C 8
front
top
side
top
front
side
front
top
side
top
front
side
69
8.3B Identifying 3-D Objects from Given 2-D Representations
To identify the shape of a 3-D object, we need to know its 2-D representations
from three different views: the front, the top and the side.
Example 2 Instant Drill 2
Sketch a 3-D object that can satisfy the
following front, top and side views.
front top side
Sol
The required 3-D object is shown below:
Sketch a 3-D object that can satisfy the
following front, top and side views.
front top side
Sol
The required 3-D object is shown below:
Step 1111: by front view,
front Step 2222: by top view,
top
Step 1111: by front view,
front Step 2222: by top view,
top
side
Step 3333: by side view,
side
Step 3333: by side view,
front
top
side
70
5. Sketch a 3-D object that can satisfy the following front,
top and side views.
front top side
Example 3 Instant Drill 3
Sketch a 3-D object that can satisfy the
following front, top and side views.
front top side
Sol
The required 3-D object is shown below:
Sketch a 3-D object that can satisfy the
following front, top and side views.
front top side
Sol
The required 3-D object is shown below:
Step 1111: by front view,
front
side
Step 3333: by side view,
Step 2222: by top view,
top
front
top
side
Step 1111: by front view,
front
side
Step 3333: by side view,
Step 2222: by top view,
top
71
6. Sketch a 3-D object that can satisfy the following front,
top and side views.
front top side
○○○○→→→→ Ex 8C 9–11
Example 4 Instant Drill 4
Sketch a 3-D object that can satisfy the given
front, top and side views of the object.
front top side
Sol The required 3-D object is shown
below:
Sketch a 3-D object that can satisfy the given
front, top and side views of the object.
front top side
Sol The required 3-D object is shown below:
In each of the following, sketch a 3-D object that can satisfy the given front, top and side views of
the object. [Nos. 7–8]
7.
front top side
8.
front top side
○○○○→→→→ Ex 8C 12–15
front
top
side
72
Level Up Questions
9. The 3-D object on the right is made up of ten identical cubes.
Draw the front, top and side views of the object.
Front view: Top view: Side view:
10. A 3-D object is made up of some identical cubes. The figures below show three of its 2-D
representations.
front top side
(a) Sketch a 3-D object that can satisfy the given information.
(b) How many cubes are there in the object?
front axis of
top
side
73
New Century Mathematics (2nd Edition) 3B
8 More about 3-D Figures
Level 1
In Nos. 1−−−−2, draw the front, top and side views of the given 3-D objects.
1. 2.
The following 3-D objects are made up of identical cubes. Draw the front, top and side views of
each of the objects. [Nos. 3–4]
3. 4.
5. The following 3-D object is made up of four identical cubes and a triangular prism. Draw the
front, top and side views.
top
front
side
top
side
front
top
side
front
top
side
front
top
side front
Consolidation Exercise
8C
74
In each of Nos. 6–8, sketch a 3-D object that can satisfy each of the following front, top and side
views of the object.
Front Top Side
6.
7.
8.
In each of Nos. 9–10, sketch a 3-D object that can satisfy the given front, top and side views of the
object.
Front Top Side
9.
10.
Level 2
In Nos. 11−−−−14, draw the front, top and side views of the given 3-D objects.
11. 12. top
side front
top
side
front
75
13. 14.
In each of Nos. 15–16, sketch a 3-D object that can satisfy the given front, top and side views of
the object.
Front Top Side
15.
16.
17. A 3-D object is made up of identical cubes. The figures below show three of its 2-D
representations.
(a) Sketch a 3-D object that can satisfy the given information.
(b) How many cubes are there in the object?
18. A 3-D object is made up of identical cubes. The figures below show three of its 2-D
representations.
(a) Sketch a 3-D object that can satisfy the given information.
(b) How many cubes are there in the object?
top
ffff sssstop
side
front
front
top
side
front
top
side
76
Answer
Consolidation Exercise 8C
17. (b) 4
18. (b) 7
77
F3B: Chapter 8D
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 10
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 11
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 12
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 13
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
8D Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8D Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8D Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
8D Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
78
Book 3B Lesson Worksheet 8D (Refer to §8.4)
8.4A Distance between a Point and a Straight Line
Refer to the figure.
PP′ is the distance between point P and the straight line ℓ .
1. The figure shows a cuboid. Find the distance between
(a) point D and line EF,
(b) point F and line HG,
(c) point H and line FG.
2. The figure shows a cuboid. Name the line segment which represents
the distance between
(a) point A and line DE,
(b) point F and line GB.
○○○○→→→→ Ex 8D 2
8.4B Relationship between Two Straight Lines
(a) In the figure, 1ℓ // 2ℓ . We say that d is the distance
between 1ℓ and 2ℓ .
(b) In the figure, 1ℓ and 2ℓ are two non-parallel straight
lines which lie in the same plane.
(i) P is the point of intersection of 1ℓ and 2ℓ .
(ii) θ (θ ≤ 90°) is the angle between 1ℓ and 2ℓ .
G
B
C
H
F
E
D
A
D
H
C
A B
G F
E 8 cm
3 cm
6 cm
3 cm F
D
E
A___ ⊥ DE
79
3. The figure shows a cuboid.
(a) Name the line segment which represents the distance between
the lines
(i) PQ and SR,
(ii) ST and RW.
(b) Name the angle between the lines
(i) PQ and PR,
(ii) PV and VQ.
4. The figure shows a right prism. Name the angle between the lines
(a) EC and DC,
(b) EB and BC,
(c) EB and AB.
○○○○→→→→ Ex 8D 3, 4
8.4C Relationship between a Point and a Plane
In the figure, P is a point outside plane π and
P′ is a point on plane π.
If PP′ is perpendicular to every line on π
which passes through P′, then:
(i) P′ is the projection of P on the plane π,
(ii) the length of PP′ is the distance between
P and the plane π.
5. The figure shows a right prism.
(a) Find the projection of
(i) point B on the plane DEF,
(ii) point C on the plane ABFE.
(b) Name the line segment which
represents the distance between
(i) point B and the plane DEF,
(ii) point C and the plane ABFE.
� PP′ ⊥ 1ℓ , PP′ ⊥ 2ℓ .
Q P
R S
Q P
U V
W
R S
T
Q P
R S
E
C B
F
A D
B
D
E F
E C
B
F
A
E
C
B
F
A
D
80
6. The figure shows a cube. Find the projection of
(a) point B on the plane ADEF,
(b) point E on the plane ABGF.
7. The figure shows a right prism with a trapezoidal base.
Find the projection of point A on the plane
(a) EFGH,
(b) CDEH.
○○○○→→→→ Ex 8D 5, 7(a)
8.4D Relationship between a Straight Line and a Plane
In the figure, straight line ℓ and plane π
intersect at point O. P is a point on the line ℓ
and Q is the projection of P on the plane π.
Then:
(i) OQ is the projection of OP on the plane π,
(ii) θ is the angle between the line ℓ and the
plane π.
e.g. The figure shows a cuboid ABCDEFGH.
(i) DE is the projection of DF on the
plane CDEH.
(ii) ∠FDE is the angle between line DF and
the plane CDEH.
G F
A B
C
H E
D M
G F
A B
C
H E
D
G B
C H
F
E D
A
projection
of OP on π
81
Example 1 Instant Drill 1
The figure shows a cube.
(a) Find the projection of CF on the plane
EFGH.
(b) Name the angle between line CF and the
plane EFGH.
Sol
(a) CF and the plane EFGH intersect at
F, and the projection of C on the plane
EFGH is H.
∴ HF is the projection of CF
on the plane EFGH.
(b) ∠CFH is the angle between line CF
and the plane EFGH.
The figure shows a right prism.
(a) Find the projection of PU on the plane
QRSU.
(b) Name the angle between line PU and the
plane QRSU.
Sol
(a) PU and the plane QRSU intersect
at , and the projection of P on
the plane QRSU is .
∴ is the projection of PU
on the plane QRSU.
(b) is the angle between line PU
and the plane QRSU.
8. Refer to the cube in Example 1.
(a) Find the projection of CF on the plane
CDEH.
(b) Name the angle between line CF and
the plane CDEH.
○○○○→→→→ Ex 8D 6, 7(b)
9. Refer to the prism in Instant Drill 1.
Name the angle between PU and each of
the following planes.
(a) PRST (b) PQR
○○○○→→→→ Ex 8D 8, 9
A D
E F
G
B C
H
P
Q R
S
T
U
CF intersects
EFGH at
F.
H is the
projection
of C on
EFGH.
HF is the
projection
of CF on
EFGH.
E F
G
C
H
E F
G
C
H
E F
G
C
H
P
Q
R
S U
P
Q
R
S U
P
Q R
S U
PU intersects
QRSU at
___.
___ is the
projection
of P on
QRSU.
___ is the
projection
of PU on
QRSU.
82
8.4E Relationship between Two Planes
In the figure, α and β are two non-parallel planes which
intersect at a straight line AB (line of intersection). Since
PQ ⊥ AB and PR ⊥ AB, we say that
∠RPQ is the angle between the planes α and β.
10. A cuboid is shown on the right. AC meets BD at X while
EG meets FH at Y. Name the lines of intersection for the
following pairs of planes.
(a) BADC and BGHC
(b) BGFA and AFHC
(c) BGED and AFHC
○○○○→→→→ Ex 8D 10
Example 2 Instant Drill 2
The figure shows a cuboid. Name the angle
between the planes ABHE and EFGH.
Sol
EH is the line of intersection of the planes
ABHE and EFGH.
∵ AE ⊥ EH and FE ⊥ EH.
∴ ∠AEF is the angle between the planes
ABHE and EFGH.
The figure shows a cube. Name the angle
between the planes VQST and WRST.
Sol
is the line of intersection of the
planes VQST and WRST.
∵ ⊥ and ⊥ .
∴ is the angle between the
planes VQST and WRST.
B
A
C
D
G
F
H
E
P U
T S
R
Q V
W
G
A
B
C H
D
G
F A
B
C H
E D
X Y
EH is the line of
intersection of
ABHE and EFGH.
∵ AE ⊥ EH and
FE ⊥ EH. ∴ ∠AEF is the angle
between the planes.
B
A H
F
G
E
A H
F
G
E
B
� ∵ BH ⊥ EH and GH ⊥ EH.
∴ ∠BHG is also the angle between
the planes.
P
A
B
Q
R
α
β
T S
R
Q V
W
T S
R
Q V
W
____ is the line of
intersection of
VQST and WRST.
∵ QS ⊥ ____ and
RS ⊥ ____. ∴ ∠______ is the angle
between the planes.
83
11.
The figure shows a cuboid. Name the
angles between the following planes.
(a) Planes ABGF and ADEF
(b) Planes ABHE and CDEH
12.
The figure shows a cuboid. M and N are
the mid-points of FE and GH respectively.
Name the angle between the planes ABNM
and MNCD.
○○○○→→→→ Ex 8D 11, 12
Level Up Question
13. The figure shows a right triangular prism. Name the angle between
(a) the lines PQ and PT,
(b) the line RU and the plane STU,
(c) the planes PQUT and PRST.
B
A
C
D
G
F
H
E
A B
D C
E
F
H
G
M N
P Q
R
T
S
U
84
C
A F
E D
B
H
G
Q
P
R S
C
A D
F
G
B
H
E
3 cm
9 cm
6 cm
A
F
C
B
D
E
New Century Mathematics (2nd Edition) 3B
8 More about 3-D Figures
Level 1
1. The figure on the right shows a cuboid.
(a) Does each of the following pairs of lines lie on the
same plane?
(i) AB, DC (ii) DE, GF
(iii) GH, CH (iv) FE, AD
(b) For those pairs of lines lying on the same plane in (a),
which of them
(i) are parallel lines? (ii) are perpendicular lines?
2. The figure on the right shows a tetrahedron. Name the line
segment which represents the distance between
(a) point P and line QS,
(b) point Q and line SR.
3. The figure shows a cuboid. Find the distance between
(a) point D and plane EFGH,
(b) point B and plane ADEF.
4. The figure shows a right triangular prism. Name the
angles between the following lines.
(a) BE and ED
(b) AC and BC
Consolidation Exercise
8D
85
P Q
R S
T
U V
X P
U
R
Q
S
T
C
A D
F
G
B
H
E
A B
C D
E
F G
H
P Q
A C
E
F
B
D
5. The figure shows a cube. Find the projection of
(a) point S on plane TUVX,
(b) point P on plane QRXV,
(c) point X on plane PQRS.
6. The figure shows a right triangular prism. Find the
projection of
(a) line PQ on plane RSTU,
(b) line RT on plane PUT.
7. The figure shows a cuboid. Name the angle between
(a) line DH and plane EFGH,
(b) line GD and plane CDEH.
8. The figure shows a cube ABCDEFGH. AE and DF intersect at P.
BH and CG intersect at Q. Name the line of intersection of the
planes
(a) ABCD and BCHG,
(b) ADEF and CDEH,
(c) CDFG and EFGH,
(d) ABHE and CDFG.
9. The figure shows a right triangular prism. Name the angle
between the planes
(a) ABFE and ACDE,
(b) DEF and ACDE,
(c) BCDF and ABFE.
86
A
C
E
B
D
G 12 cm
8 cm
16 cm 10 cm
6 cm
F
H
P
Q R
U S
T
X
P
Q
R
U
S
T
F
E
D
A
C
B
Level 2
10. The figure shows a right prism. Find the distance between
(a) point D and line AF,
(b) point G and plane ABCD,
(c) lines EH and FG.
11. The figure shows a cuboid. Find the projection of
(a) line BG on the plane CDEH,
(b) line AH on the plane ADEF,
(c) line EG on the plane ABCD.
12. The figure shows a right triangular prism PQRSTU.
X is a point on QR such that PX ⊥ QR. Name the angle
between line PS and
(a) plane QRSU,
(b) plane PQUT.
13. The figure shows a right hexagonal prism. It is given
that FD ⊥ DC and the plane AFRS is a rectangle.
(a) Name the angle between line FS and plane
(i) ABCDEF,
(ii) CDRS.
(b) Name the angle between planes
(i) FDR and DEQR,
(ii) AFRS and CDRS.
87
14. The figure shows a tetrahedron PQRS, where PQ = PR, QS = RS, PS ⊥ QS and PS ⊥ SR. N is
the mid-point of QR.
Name the angle between planes
(a) PQS and PRS,
(b) PQR and SQR.
15. The figure shows a right prism ABCDEF. Its base is an equilateral triangle. X is the mid-point of
DF.
(a) Amy claims that the angle between line AF and plane CBFD is ∠AFB. Do you agree?
Explain your answer.
(b) Benson claims that the angle between planes AFD and EFD is ∠AXE. Do you agree?
Explain your answer.
P
Q
R
S
N
A
B
C
E
F
D
X
88
Answer
Consolidation Exercise 8D
1. (a) (i) yes (ii) no
(iii) yes (iv) yes
(b) (i) AB, DC; FE, AD
(ii) GH, CH
2. (a) PS (b) QS
3. (a) 3 cm (b) 9 cm
4. (a) ∠BED (b) ∠ACB
5. (a) T (b) Q (c) R
6. (a) UR (b) UT
7. (a) ∠DHE (b) ∠GDH
8. (a) BC (b) DE
(c) FG (d) PQ
9. (a) ∠BAC or ∠FED
(b) ∠FEA
(c) ∠CBA or ∠DFE
10. (a) 10 cm (b) 16 cm (c) 8 cm
11. (a) CH (b) AE (c) DB
12. (a) ∠PSX (b) ∠SPT
13. (a) (i) ∠SFC (ii) ∠FSD
(b) (i) ∠FDE (ii) ∠FRD or ∠ASC
14. (a) ∠QSR (b) ∠PNS
15. (a) no (b) yes
89
F3B: Chapter 9A
Date Task Progress
Lesson Worksheet
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(Full Solution)
Book Example 1
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 2
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 3
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 4
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 5
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
9A Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9A Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9A Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9A Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
90
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
91
Book 3B Lesson Worksheet 9A (Refer to §9.1)
9.1 Pyramids
(a) If a solid has a polygonal base and all its other faces
are triangles with a common vertex, then the solid is
a pyramid.
(b) The figure shows a right pyramid with a rectangular
base. It has the following properties:
(i) NA = NB = NC = ND
(ii) VA = VB = VC = VD
(iii) △VAB, △VBC, △VCD and △VDA are
isosceles triangles.
Note: If the base of a right pyramid is a regular polygon, this right pyramid is called a
regular pyramid.
9.1A Volume of a Pyramid
Volume of a pyramid =
3
1× base area × height
Example 1 Instant Drill 1
Find the volume of the
triangular pyramid in
the figure.
Find the volume of the
rectangular pyramid in
the figure.
Sol Volume =3
1× 18 × 10 cm3
= 60 cm3
Sol Volume =3
1× ( ) × ( ) cm3
=
○○○○→→→→ Ex 9A 1
height
= 10 cm
base area
= 18 cm2
height
= 6 cm
base area
= 25 cm2
base area
height
92
1. Find the volume of the pyramid with a
square base in the figure.
2. Find the volume of the rectangular
pyramid in the figure.
3. Find the volume of the triangular pyramid
in the figure.
4. The figure shows a regular pyramid with
volume 189 m3. Find the height of the
pyramid.
Let h m be the height of the pyramid.
○○○○→→→→ Ex 9A 2−6 ○○○○→→→→ Ex 9A 9, 11, 12
8 cm
3 cm
7 m
6 m
5 m
13 cm
6 cm 9 cm 9 m
What is the
base
area?
93
9.1B Volume of the Frustum of a Pyramid
(a) When a pyramid is cut by a plane parallel to its base, the removed part at the top is a
small pyramid and the remaining part is called the frustum of the pyramid.
(b) Volume of a frustum of a pyramid
= volume of the larger pyramid − volume of the smaller pyramid
Example 2 Instant Drill 2
The upper base and the lower base of the
frustum ABCDEFGH in the figure are squares.
Find the volume of the frustum.
The upper base and the lower base of the
frustum ABCDEFGH in the figure are squares.
Find the volume of the frustum.
Sol
Volume of the frustum
= volume of pyramid VABCD −
volume of pyramid VEFGH
=
××−+×× 35
3
1)63(15
3
1 22 cm3
= [675 − 25] cm3
= 650 cm3
Sol Volume of the frustum
= volume of pyramid VABCD −
volume of pyramid __________
=
15 cm A B
C
G
E F
H
D
V
5 cm
3 cm
6 cm 5 m
13.5 m
4 m
6 m
A B
C
G E
F
H
D
V
=
−
94
5. The frustum shown in the figure is formed by cutting
the small right pyramid VEFGH from the large right
pyramid VABCD. The bases of the two right pyramids
are rectangles. Find the volume of the frustum.
○○○○→→→→ Ex 9A 15, 16
10 cm
9 cm A
B
C
G
E
F H
D
V
6 cm
2 cm
3 cm
5 cm
95
9.1C Total Surface Area of a Pyramid
Total surface area of a pyramid
= total area of all lateral faces + area of the base
Example 3 Instant Drill 3
In the figure, VABCD is a right pyramid with a
square base. Find the total surface area of the
pyramid.
In the figure, VPQRS is a right pyramid with a
square base. Find the total surface area of the
pyramid.
Sol Total surface area of the pyramid
= 4 × area of △VBC +
area of square ABCD
=
+××× 2101210
2
14 cm2
= 340 cm2
Sol
6. The figure shows a regular pyramid
VABCD. Find the total surface area of the
pyramid.
7. In the figure, the base of the right pyramid
VABCD is a rectangle. Find the total
surface area of the pyramid.
○○○○→→→→ Ex 9A 7, 8, 14
Level Up Questions
8. In the figure, VABCD is a right pyramid with a square base.
Its total surface area is 264 cm2. Find the length of a side of
its base.
V
V
V
V
A B
C D lateral
fac
lateral
fac
base
10 cm A
B
C D
V 12 cm
E
15 cm A
B
C D
V 17 cm
E
4 × +
A B
C D
V
20 m
F
13 m
E
32 m
10 m
2 × + 2 × +
Set up an equation
for total surface
area.
15 cm
7 cm
T Q
R P
V S
B
C A
V D
area of the lateral face = 50 cm2
96
9. In the figure, VABCD is a right rectangular pyramid. Find
(a) EC,
(b) VE,
(c) the volume of the pyramid.
(Give the answers correct to 3 significant figures if
necessary.)
(a) In the right-angled triangle ABC,
AC = 22)()( + �Pyth. theorem
=
(b) In the right-angled triangle VEC,
(c)
A
B
C D
V
8 m 15 m
E
13 m
The diagonals of a rectangle
bisect each other. So, EC
=2
1× AC.
97
New Century Mathematics (2nd Edition) 3B
9 Area and Volume (III)
Level 1
Find the volumes of the following pyramids. [Nos. 1−−−−5]
1. 2. 3.
4. 5.
Find the total surface area of each of the following regular pyramids. [Nos. 6−−−−7]
6. 7.
7 cm
base area = 24 cm2
6 cm
4 cm
5 cm
36 m
The base is a square.
10 m
5 mm
12 mm
9 mm
9 cm
9 cm
area of the
lateral
face
= 60
25 cm
16 cm
16 cm
Consolidation Exercise
9A
98
4 cm 4 cm
total surface area = 56 cm2
h cm
Find the unknown in each of the following right pyramids. [Nos. 8−−−−9]
8. 9.
10. The figure shows a pyramid. The length of each side of the square base is 5 cm. If the volume of
the pyramid is 60 cm3, find the height of the pyramid.
11. The figure shows a pyramid PQRS where RS = 6 cm and the height PQ = 7 cm.
If the volume of the pyramid is 28 cm3,
(a) find the area of △QRS,
(b) find the lengths of QR and the slant edge PR.
(Give the answers correct to the nearest 0.1 cm if necessary.)
base area = x m2
20 m
volume = 800 m3
5 P
Q
R
S
7 cm
6 cm
99
12. The figure shows a right pyramid ABCDE, where BC = 30 cm and CD = 14 cm. The heights of
the lateral faces △ABC and △ACD are 20 cm and 24 cm respectively. Find the total surface
area of the pyramid.
13. The frustum shown in the figure is formed by cutting pyramid VPQRS from pyramid VTUXY.
The bases of the two pyramids are rectangles. Find the volume of the frustum.
14. The frustum shown in the figure is formed by cutting pyramid ABCDE from pyramid AFGHI.
The bases of the two pyramids are squares. Find the volume of the frustum.
30 cm
24 cm
20 cm
14 cm
A
B
C
D
E
10 cm
15 cm
6 cm
9 cm
18 cm
V
P
Q R
S
X U
T Y
12 cm
100
12 cm 16 cm
26 cm
A
B
C
D
E
N
Level 2
15. The figure shows a pyramid VPQR whose height is VP. It is given that VQ = 7.5 m, PQ = 6 m
and QR = 10 m. Find the volume of the pyramid.
16. The figure shows a right pyramid ABCDE with a rectangular
base BCDE. The diagonals BD and CE intersect at N. AD = 26 cm,
BC = 12 cm and CD = 16 cm.
(a) Find the lengths of ND and AN.
(b) Find the volume of the pyramid.
17. The cardboard in Fig. A is the net of the right square pyramid VPQRS in Fig. B. It is given that
VN = 30 cm and VQ = 34 cm.
(a) Find the length of PQ.
(b) Find the total surface area of the pyramid.
10 m
V
P
Q
R 7.5 6 m
Fig. A Fig. B
V
R
Q
P
S NV
R Q
P
V
S
N
34 cm
30 cm
V
V
101
A
B
C
D
E
60 cm
50 cm
18. The figure shows a food tent without base. It is
in the shape of a right square pyramid ABCDE,
where BC = 60 cm and AB = 50 cm. Find the
total surface area of the tent.
19. In the figure, VPQRS is a right pyramid whose base PQRS is a rectangle. The height VX of the
lateral face △VPQ is 6.8 cm. PQ = 10.2 cm and QR = 8 cm.
(a) Find the length of VQ.
(b) Find the height of the lateral face △VQR.
(c) Find the total surface area of the pyramid.
20. The figure shows a regular pyramid VABCD. It is given that the area of △VAB is 1 040 cm2 and
the total surface area of the pyramid is 5 760 cm2.
Find
(a) the area of the base ABCD,
(b) the height VQ,
(c) the volume of the pyramid.
10.2 cm
8 cm
6.8 cm
V
Q
R
S
P
X
A
B
C
V
D
Q P
102
21. The figure shows a frustum BCDEFGHI of a right pyramid. Its upper base and lower base are
rectangles with dimensions 10 cm × 7.5 cm and 16 cm × 12 cm respectively. The height of the
frustum is 9 cm. Let h cm be the height of the pyramid AFGHI.
(a) Find the value of h.
(b) Find the volume of the frustum.
22. The figure shows a vessel of capacity 324 cm3. Its shape is an inverted right pyramid whose
base is a horizontal square XYTU of side 9 cm. The vessel contains water to the depth of 8 cm.
(a) Find the height of the vessel.
(b) Find the additional volume of water required to fill up the vessel.
9 cm
16 cm
A
B
C D
E
H I
F G
h cm
12 cm
10 cm
7.5 cm
V
P
Q
R
T
S
U
X
Y
8 cm
9 cm
103
Answer
Consolidation Exercise 9A
1. 56 cm3
2. 40 cm3
3. 1 200 m3
4. 90 mm3
5. 800 cm3
6. 321 cm2
7. 1 056 cm2
8. 120
9. 5
10. 7.2 cm
11. (a) 12 cm2
(b) QR = 4 cm, PR = 8.1 cm
12. 1 356 cm2
13. 570 cm3
14. 148 m3
15. 45 m3
16. (a) ND = 10 cm, AN = 24 cm
(b) 1 536 cm3
17. (a) 32 cm (b) 2 944 cm2
18. 4 800 cm2
19. (a) 8.5 cm (b) 7.5 cm
(c) 210.96 cm2
20. (a) 1 600 cm2 (b) 48 cm
(c) 25 600 cm3
21. (a) 24 (b) 1 161 cm3
22. (a) 12 cm (b) 228 cm3
104
F3B: Chapter 9B
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 6
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 7
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 8
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 9
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 10
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
9B Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9B Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9B Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9B Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
105
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
106
Book 3B Lesson Worksheet 9B (Refer to §9.2)
9.2 Circular Cones
(a) A circular cone is a solid enclosed by a circular
base and a curved surface.
(b) (i) If all the slant heights of a circular cone are
equal, the cone is called a right circular
cone.
(ii) By Pythagoras’ theorem,
2ℓ = r2 + h2.
9.2A Volume of a Right Circular Cone
Volume of a circular cone =3
1πr2h
Example 1 Instant Drill 1 Find the volume of the right circular cone in
the figure in terms of π.
Find the volume of the right circular cone in
the figure in terms of π.
Sol Volume =3
1× π × 32 × 6 cm3
= 18π cm3
Sol Volume =3
1× π × ( ) × ( ) cm3
=
3 cm
6 cm
7 cm
9 cm
3
1× base area ×
height
height h slant
height
ℓ
base radius r
height h base radius r
107
1. Find the volume of the right circular cone
in the figure in terms of π.
2. The volume of the right circular cone in
the figure is 940 cm3. Find its height.
( Give the answer correct to 3 significant
figures.)
○○○○→→→→ Ex 9B 1, 2 ○○○○→→→→ Ex 9B 4
Example 2 Instant Drill 2 For the following right circular cone, find
(a) the height,
(b) the volume in terms of π.
For the following right circular cone, find
(a) the height,
(b) the volume in terms of π.
Sol (a) Let h cm be the height
of the cone.
( )2 + ( )2 = ( )2
=
∴ The height of the cone is .
(b) Volume
=3
1× ( ) × ( ) × ( ) cm3
=
Sol (a) Let h cm be the height of the cone.
82 + h2 = 172
h = 22 817 −
= 15
∴ The height of the cone is 15 cm.
(b) Volume =3
1× π × 82 × 15 cm3
= 320π cm3
6 cm
11 cm
10 cm
8 cm
17 cm
20 cm
16 cm
8 cm
17 cm h cm
16 cm
20 cm h cm
108
3. For the following right circular cone, find
(a) the base radius,
(b) the volume, correct to 3 significant
figures.
4. For the following right circular cone, find
(a) the height,
(b) the volume, correct to 3 significant
figures.
○○○○→→→→ Ex 9B 3
9.2B Volume of the Frustum of a Circular Cone
(a) When a right circular cone is cut by a plane parallel to its base, the removed part at
the top is a small right circular cone and the remaining part is called the frustum of
the circular cone.
(b) Volume of a frustum of a circular cone
= volume of larger circular cone − volume of smaller circular cone
12 cm
13 cm 37 cm
24 cm
109
Example 3 Instant Drill 3
Find the volume of the frustum of the right
circular cone below in terms of π.
Find the volume of the frustum of the right
circular cone below in terms of π.
Sol
Volume of the frustum
= volume of circular cone VAB −
volume of circular cone VCD
=
×××−××× 43π
3
186π
3
1 22 cm3
= (96π − 12π) cm3
= 84π cm3
Sol Volume of the frustum
= volume of circular cone ______ −
volume of circular cone ______
=
5. Find the volume of the frustum of the right circular cone below in terms of π.
○○○○→→→→ Ex 9B 14, 15
6 cm
8 cm 3 cm
V
A B
C D
4 cm
9 cm
5 cm 3 cm
P Q
R S
V
10 cm
= −
20 cm
9 cm 15 cm
A B
C D
V
27 cm
= −
110
9.2C Total Surface Area of a Right Circular Cone
(a) Curved surface area of a right circular cone
= ℓrπ
(b) Total surface area of a right circular cone
= ℓrπ + πr2
Example 4 Instant Drill 4
Find
(a) the curved surface area, and
(b) the total surface area
of the following right circular cone.
(Give the answers in terms of π.)
Find
(a) the curved surface area, and
(b) the total surface area
of the following right circular cone.
(Give the answers in terms of π.)
Sol (a) Curved surface area
= π × 5 × 13 cm2
= 65π cm2
(b) Total surface area
= (65π + π × 52) cm2
= 90π cm2
Sol (a) Curved surface area
= π × ( ) × ( ) cm2
=
(b) Total surface area
=
13 cm
5 cm
20 cm
12 cm
ℓ ℓ
ℓ
111
6. Find
(a) the curved surface area, and
(b) the total surface area
of the following right circular cone.
(Give the answers in terms of π.)
7. Find
(a) the curved surface area, and
(b) the total surface area
of the following right circular cone.
(Give the answers in terms of π.)
○○○○→→→→ Ex 9B 5, 6
8. In the figure, the curved surface area of a
right circular cone is 24π m2 and the slant
height is 8 m. Find
(a) the base radius, and
(b) the total surface area
of the circular cone.
(Give the answers correct
to 3 significant figures if neccesary.)
9. In the figure, the total
surface area of a right
circular cone is 42π cm2
and the base radius is 3 cm.
Find the slant height of the
circular cone.
Let ℓ cm be the slant height.
(a) Let r m be the base radius.
π × ( ) × ( ) = ( )
=
(b) Total surface area
=
○○○○→→→→ Ex 9B 7, 8, 12
7 cm
16 cm 10 cm
13 cm
3 cm
8 m
112
���� ‘Explain Your Answer’ Question
10. Mary pours 500 cm3 of water into an empty container. The container is in the shape of a right
circular cone of base radius 6 cm and height 13 cm. Will the water overflow? Explain your
answer
Capacity of the container =
∴ The water (will / will not) overflow.
Level Up Questions
11. In the figure, the volume of a right circular cone is 800π cm3 and
its base radius is 10 cm.
(a) Find the height of the circular cone.
(b) Find the total surface area of the circular cone in terms of π.
(a) Let h cm be the height.
(b) Let ℓ cm be the slant height.
12. It is known that the curved surface area of a right circular cone is 369π cm2 and its slant height
is 41 cm.
(a) Find the base radius of the cone.
(b) Find the volume of the cone in terms of π.
Find the slant
height
first.
10 cm
113
h
r
New Century Mathematics (2nd Edition) 3B
9 Area and Volume (III)
Level 1
Find the volume of each of the following right circular cones in terms of π. [Nos. 1−−−−3]
1. 2. 3.
4. The figure shows a right circular cone of base radius r and height h.
Complete the following table.
r h Volume
(a) 4 m 32π m3
(b) 5 cm 15π cm3
For each right circular cone in Nos. 5–7, find
(a) the curved surface area,
(b) the total surface area.
(Give the answers in terms of π.)
5. 6. 7.
9 mm
5 mm
45 cm
16 cm
4.5 m 7.5 m
4 cm
8 cm
20
m
33 m
3.6 mm
1.5 mm
Consolidation Exercise
9B
114
ℓ
r
8. The figure shows a right circular cone of base radius r and slant height ℓ .
Complete the following tables.
r ℓ Total surface area
(b) (i) 5 mm 80π mm2
(ii) 7 cm 112π cm2
9. The figure shows a tent in the shape of a right circular cone. The curved surface and the base are
made of the same material. The slant height is 2.5 m and the base area is 4π m2.
(a) Find the base radius of the tent.
(b) Find the area of the material used to make the curved surface of the tent in terms of π.
10. The figure shows a right circular cone of height 7.5 cm. If the circumference of its base is
8π cm, find the volume of the cone in terms of π.
11. The base radius and height of a right circular cone are 7 cm and 11 cm respectively. Find the
total surface area of the cone.
(Give the answer correct to the nearest cm2.)
r ℓ Curved surface area
(a) (i) 3 m 12π m2
(ii) 14 cm 70π cm2
2.5 m
7.
115
12. Lucy has eight identical cups in the shape of inverted right circular cones. The base diameter
and the height of each cup are 9 cm and 14 cm respectively. Can the eight cups hold orange
juice of volume 2 250 cm3? Explain your answer.
13. The figure shows a pudding in the shape of the frustum of a right circular cone. The radii of its
upper base and lower base are 6 cm and 7.5 cm respectively. AP = 24 cm and PQ = 6 cm. Find
the volume of the pudding correct to the nearest cm3.
Level 2
14. In the figure, the volume of a right circular cone is 96π cm3 and its height is 8 cm.
(a) Find the base radius and the slant height of the circular cone.
(b) Find the total surface area of the circular cone in terms of π.
15. The figure shows an inverted right circular cone. Its base diameter and curved surface area are
32 cm and 544π cm2 respectively.
(a) Find the height of the circular cone.
(b) Find the volume of the circular cone in terms of π.
7.5 cm
6 cm
24 cm
6 cm
A
P
Q
B C
D E
8 cm
32 cm
116
16. Fig. A shows a glass filled up with apple juice. It is in the shape of an inverted right circular
cone of base radius 9 cm and height 15 cm. All the juice in the glass is poured into an empty
cylindrical vessel of base radius 6 cm (as shown in Fig. B). Find the depth of the juice in the
cylindrical vessel.
17. In the figure, the dimensions of a chocolate cuboid are 12.5 cm × 4 cm × 4 cm. It is melted and
recast into some identical right circular cones of base radius 1.5 cm and height 2.4 cm. What is
the maximum number of cones that can be made?
18. In the figure, a storage tank is composed of a right circular cone and a cylinder with a common
base. The height of the tank is 4 m. The slant height of the cone is 3.25 m and the base diameter
of the cylinder is 6 m.
(a) Find the capacity of the tank.
(b) Find the total curved surface area of the tank.
(Give the answers in terms of π.)
15 cm
9 cm 6 cm
Fig.
Fig.
4 cm
4 cm 12.5 cm
3.25 m
6 m
4 m
117
19.
The figure shows a bowl. Its shape is a frustum of an inverted right circular cone. The radii of
its upper base and lower base are 10 cm and 7.5 cm respectively. Find the capacity of the bowl
in terms of π.
20.
Fig. A shows a paper cup in the shape of an inverted right circular cone.
(a) Find the capacity of the paper cup.
(b) The paper cup is cut along PQ to form the sector in Fig. B. Find
(i) the area of the sector,
(ii) the angle of the sector.
(Give the answers correct to 3 significant figures.)
48 cm
A
C B
E D
P
Q 10 cm
7.5 cm
20.5 cm 20 cm
Q
P
Q
P
Fig. A Fig. B
O
118
21.
Fig. A shows a sector. It is folded into an inverted right circular cone as shown in Fig. B.
(a) Find the base radius of the cone.
(b) Find the capacity of the cone.
(Give the answer correct to 3 significant figures.)
22.
The figure shows a vessel in the shape of an inverted right circular cone. The vessel contains
some water. The area of the surface of the vessel in contact with water is 252π cm2.
(a) Find the depth of water in the vessel.
(b) Find the additional volume of water required to fill up the vessel.
(Give the answer correct to the nearest cm3.)
12 cm
Fig. B Fig. A
240°
10.5 cm
8.4 cm
O B
A
C
D E P
119
Answer
Consolidation Exercise 9B
1. 75π mm3
2. 960π cm3
3. 54π cm3
4. (a) 6 m (b) 3 cm
5. (a) 32π cm2 (b) 48π cm2
6. (a) 330π m2 (b) 430π m2
7. (a) 5.85π mm2 (b) 8.1π mm2
8. (a) (i) 4 m (ii) 5 cm
(b) (i) 11 mm (ii) 9 cm
9. (a) 2 m (b) 5π m2
10. 40π cm3
11. 441 cm2
12. yes
13. 862 cm3
14. (a) base radius = 6 cm, slant height = 10 cm
(b) 96π cm2
15. (a) 30 cm (b) 2 560π cm3
16. 11.25 cm
17. 35
18. (a) 28.5π m3 (b) 26.25π m2
19. 925π cm3
20. (a) 424 cm3
(b) (i) 290 cm2 (ii) 79.0°
21. (a) 8 cm (b) 599 cm3
22. (a) 28.8 cm (b) 2 028 cm3
120
F3B: Chapter 9C
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 11
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 12
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 13
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 14
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
9C Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9C Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9C Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9C Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
121
Book 3B Lesson Worksheet 9C (Refer to §9.3)
9.3 Spheres
(a) The figure shows a sphere.
(i) The point O is called the centre.
(ii) The distance between the centre O and any
point on the surface is the radius (r) of the
sphere.
(b) When a sphere is cut along a plane which passes
through the centre, it is divided into two identical
parts called hemispheres.
9.3A Volume of a Sphere
Volume of a sphere =3
4πr3
Example 1 Instant Drill 1
Find the volume of the sphere in the figure in
terms of π.
Find the volume of the sphere in the figure in
terms of π.
Sol Volume =3
4× π × 63 cm3
= 288π cm3
Sol Volume =3
4× π × ( )3 m3
=
6 cm 1.5 m
122
1. Find the volume of the sphere in the
figure, correct to the nearest cm3.
2. If the volume of a sphere is 36π cm3, find
its radius.
Let r cm be its radius.
3
4× π × ( )3 = ( )
=
○○○○→→→→ Ex 9C 1(a), 2(a), 3(a) ○○○○→→→→ Ex 9C 4(b), 6
3. The solid in the figure consists of a
cylinder and a sphere. Both of the base
radius of the cylinder and the radius of the
sphere are 6 cm. Find the volume of the
solid in terms of π.
Volume of the solid
= volume of the sphere +
volume of the cylinder
=
4. The solid in the figure consists of a
hemisphere and an inverted right circular
cone with a common base. The radius of
the hemisphere is 12 cm and the height of
the cone is 29 cm. Find the volume of the
solid in terms of π.
Volume of the solid
= volume of the hemisphere +
volume of the cone
=
○○○○→→→→ Ex 9C 11, 12
7 cm
6 cm 10 cm
6 cm
12 cm
29 cm
Volume of a
cylinder =
πr2h
Volume of a hemisphere
=2
1× volume of a sphere
=3
4
2
1× πr3
Volume of a cone
=3
1πr2h
123
9.3B Surface Area of a Sphere
Surface area of a sphere = 4πr2
Example 2 Instant Drill 2
Find the surface area of the sphere below in
terms of π.
Find the surface area of the sphere below in
terms of π.
Sol Surface area of the sphere
= 4 × π × 52 cm2
= 100π cm2
Sol Surface area of the sphere
= ( ) × ( ) × ( ) m2
=
5. Find the surface area of the sphere below
in terms of π.
6. The surface area of a sphere is 324π cm2.
Find its radius.
○○○○→→→→ Ex 9C 1(b), 2(b), 8
○○○○→→→→ Ex 9C 4(a), 10
5 cm 7 m
16 cm
124
7. Find the total surface area of the
hemisphere below in terms of π.
8. Find the total surface area of the
hemisphere below, correct to the nearest
cm2.
○○○○→→→→ Ex 9C 3(b)
9. The total surface area of a hemisphere is
300π mm2. Find its radius.
10. The curved surface area of a hemisphere is
242π m2.
(a) Find its radius.
(b) Find its volume, correct to
3 significant figures.
○○○○→→→→ Ex 9C 5, 14
6 cm
15 cm
+
curved surface circular
plane
125
Level Up Questions
11. Refer to the figure. A solid metal sphere of radius 6 cm is melted and recast to form a solid
right circular cone of height 13.5 cm. Find the base radius of the circular cone.
Let r cm be the base radius of the circular cone.
Volume of the sphere =
Volume of the circular cone =
∵ Volume of the sphere (= / ≠) volume of the circular cone
∴
12. It is known that the volume of a sphere is 36π cm3. Find the surface area of the sphere, correct
to 3 significant figures.
6 cm 13.5 cm
Suppose the volume of metal
does not change during the
process.
You may keep the
π in the results.
126
New Century Mathematics (2nd Edition) 3B
9 Area and Volume (III)
Level 1
For each sphere or hemisphere in Nos. 1−−−−3, find the
(a) volume,
(b) total surface area.
(Give the answers in terms of π.)
1. 2. 3.
4. Find r in each of the following spheres.
(a) (b)
5. Find r in each of the following hemispheres.
(a) (b)
6. The volume of a sphere is 3cm π
3
500. Find its diameter.
7. The figure shows a hemisphere. The area of its base is 900π mm2.
Find its volume in terms of π.
8. The diameter of a spherical Christmas ball is 7 cm. Find the surface area of the ball in
terms of π.
9. The curved surface area of a hemisphere is 200 m2. Find the circumference of its base, correct
to 3 significant figures.
6 cm
18 mm
3 m
r cm
surface area = 100π cm2
r m
volume =3m π
3
4
r mm
curved surface area = 98π mm2
r cm
total surface area = 12π cm2
Consolidation Exercise
9C
127
10. The figure shows a toy. It consists of a right circular cone and a hemisphere with a common
base. Find the total surface area of the toy.
(Give the answer in terms of π.)
11. The surface area of a spherical basketball is 1 850 cm2. Find
(a) its radius,
(b) its volume.
(Give the answers correct to 3 significant figures.)
Level 2
12. The volumes of the sphere and the cylinder in the figure are the same. Find the base radius of
the cylinder.
13. In the figure, there is some water in a cylindrical container of base radius 4.8 cm. An iron
sphere of diameter 7.2 cm is immersed in water. Then, the depth of water rises to 13 cm. Find
the original depth of water.
14. A large solid metal sphere of radius 15 cm is melted and recast into 27 identical small solid
metal spheres. Find the surface area of each small sphere in terms of π.
6 cm 14 cm
9 cm
6.75 cm
13 cm
4.8 cm
128
15. The figure shows a steel bowl in the shape of a hemisphere. The bowl is 0.2 cm thick with an
internal diameter of 15.6 cm. If there is 600 cm3 of steel, at most how many bowls can be made?
16. The solid in the figure is formed by two hemispheres. The radius of the smaller hemisphere is
4 cm. The curved surface area of the larger hemisphere is 40.5π cm2. Find the total surface area
of the solid, correct to the nearest cm2.
17. The figure shows a right circular cone of base radius 14.4 cm and a hemisphere of radius
12 cm. Their total surface areas are the same.
(a) Find the slant height of the cone. (b) Is the volume of the hemisphere greater than that of the cone? Explain your answer.
18. In the figure, a vessel consists of a hemisphere and a cylinder with a common base. The base
diameter of the hemisphere is 30 cm. The capacity of the vessel is 9 000π cm3. Is the height of
the vessel less than 40 cm? Explain your answer.
0.2 cm
15.6 cm
4 cm
12 cm 14.4 cm
129
19. The solid in the figure consists of a right circular cone and a hemisphere with a common base.
The height of the cone is 24 cm. The ratio of the volume of the cone to the volume of the
hemisphere is 2 : 3.
(a) Find the radius of the hemisphere.
(b) Find the volume of the solid in terms of π.
(c) Find the total surface area of the solid in terms of π.
24 cm
130
Answer
Consolidation Exercise 9C
1. (a) 288π cm3 (b) 144π cm2
2. (a) 972π mm3 (b) 324π mm2
3. (a) 18π m3 (b) 27π m2
4. (a) 5 (b) 1
5. (a) 7 (b) 2
6. 10 cm
7. 18 000π mm3
8. 49π cm2
9. 35.4 m
10. 132π cm2
11. (a) 12.1 cm (b) 7 480 cm3
12. 12 cm
13. 10.3 cm
14. 100π cm2
15. 7
16. 241 cm2
17. (a) 15.6 cm (b) yes
18. no
19. (a) 18 cm (b) 6 480π cm3
(c) 1 188π cm2
131
F3B: Chapter 9D
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
9D Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9D Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9D Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9D Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
132
Book 3B Lesson Worksheet 9D (Refer to §9.4)
9.4 Dimensions of Length, Area and Volume
Measurement Dimension
Example:
Cube
Linear (involving the lengths of
line segments or curves, the sum
or multiple of lengths)
1 Perimeter of a face = 4x
Quadratic (involving the product
of two linear measurements) 2 Total surface area = 6x2
Cubic (involving the product of
three linear measurements) 3 Volume = x3
Consider the rectangular pyramid in the figure. For each expression in the table, indicate its kind of
measurement and write down its dimension. [Nos. 1– 4]
Expression Linear Quadratic Cubic Dimension
1. h
2. ab
3. abh
4. 2(a + b) ○○○○→→→→ Ex 9D 1−3
The formulae of two quantities in the following solids are given. Determine whether the quantity
represented by each formula can be the total surface area or the volume of the solid. [Nos. 5–6]
5. Regular pyramid
Formula (a): X =3
1a2h
Formula (b): Y = a2 + 2ab
x x
x
a
b
h
Degree of a1b1
= 1 + 1 =
( )
� Degree of
expression = 1
� Degree of
expression = 2
� Degree of
expression = 3
Degree of
a1b1h1
=
Find the degree of
the expression
first.
h
a
b
133
6. Cylinder
Formula (a): X = 2πr(r + h)
Formula (b): Y = πr2h
○○○○→→→→ Ex 9D 4, 5
Level Up Questions
Refer to the solids as shown. Identify which measurement (linear, quadratic or cubic) is represented
by each of the expressions below. [Nos. 7–8]
7. (a) h + r
(b) 3πr2 + 2πrh
(c) πr2h +3
2πr3
8. (a) Ah
(b) 2(x + h)
(c) 2A + 4xh
r h
h
x
area = A
r h
Note that A represents
a quadratic
measurement.
134
a
b c
x
x y
d
z
r
ℓ
a
b x
y h
New Century Mathematics (2nd Edition) 3B
9 Area and Volume (III)
Level 1
Refer to the solids as shown. Write down the dimensions of the quantities represented by the
expressions below. [Nos. 1–3]
1. (a) a + b + c
(b) bc
(c) abc
2. (a) yx2
2
1
(b) 2(x + y)
(c) yxx )22( +
3. (a) zdπ
(b) 2
d
(c) zdd23
π4
1π
12
1+
The formulae of two quantities in the following solids are given. Determine whether the quantity
represented by each formula can be the total surface area or the volume of the solid. [Nos. 4–5]
4. Right rectangular pyramid
Formula (a): abhP3
1=
Formula (b): Q = ax + ab + by
5. Half-cylinder
Formula (a): ℓrrX )2π(π2 ++=
Formula (b): ℓ2
π2
1rY =
Consolidation Exercise
9D
135
h
a
a
b
c
A
Level 2 Refer to the solids as shown. Identify which measurement (linear, quadratic or cubic) is represented
by each of the expressions below. [Nos. 6–7]
6. (a) 4a
(b) ha2
3
1
(c) 224 aha +
7. (a) Ab
(b) 22 bc −
(c) 2262 bcbA −+
8. Let a, b and c be the linear measurements of a solid. Carman and Dickson express the volume of
the solid in terms of a, b and c as a2b + bc2 – c3 and 4ab + 3bc + 2ac respectively. It is known
that one of the expressions must be wrong. Whose expression is it? Explain your answer.
(A refers to the base area of the prism.)
136
Answer
Consolidation Exercise 9D
1. (a) 1 (b) 2 (c) 3
2. (a) 3 (b) 1 (c) 2
3. (a) 2 (b) 1 (c) 3
4. (a) volume
(b) total surface area
5. (a) total surface area
(b) volume
6. (a) linear measurement
(b) cubic measurement
(c) quadratic measurement
7. (a) cubic measurement
(b) linear measurement
(c) quadratic measurement
8. Dickson
137
F3B: Chapter 9E
Date Task Progress
Lesson Worksheet
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(Full Solution)
Book Example 15
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 16
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 17
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 18
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 19
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 20
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise
9E Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9E Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
Maths Corner Exercise
9E Level 3
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s
Signature
___________
( )
138
Maths Corner Exercise
9E Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s
Signature
___________
( )
E-Class Multiple Choice
Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
139
Book 3B Lesson Worksheet 9E (Refer to §9.5)
9.5A Similar Plane Figures
For two similar plane figures,
if A1 and A2 denote their areas,1ℓ and
2ℓ denote any two corresponding lengths,
then
2
2
1
2
1
=ℓ
ℓ
A
A.
Example 1 Instant Drill 1
The figure shows two similar triangles A and
B. Find the ratio of the area of A to that of B.
The figure shows two similar rectangles P and
Q. Find the ratio of the area of P to that of Q.
Sol B
A
of Area
of Area =
2
cm6
cm10
=
2
3
5
=9
25
Sol Q
P
of Area
of Area =
)(
cm)(
cm)(
=
1. The figure shows two similar hexagons R
and S. Find the ratio of the area of R to that
of S.
2. Figures M and N as shown below are
similar. Find the ratio of the area of M to
that of N.
○○○○→→→→ Ex 9E 1, 2
6 cm 10 cm
A B 6 cm 3 cm P Q
9.6 cm 8 cm
R S
7 cm 14 cm
M N
140
3. The ratio of a pair of corresponding sides
of the two similar triangles as shown
below is 2 : 3. If the area of the large
triangle is 162 cm2, find the area of the
small triangle.
Let x cm2 be the area of the small triangle.
)(
2
2
)(
2
cm )(
cm
=
x
=
4. Figures C and D as shown below are
similar. If the area of figure C is 176 cm2,
find the area of figure D.
○○○○→→→→ Ex 9E 3−6
Example 2 Instant Drill 2
The actual area of a playground is 1 000 m2,
and its area on a plan is 2.5 cm2. If the scale of
the plan is 1 : k, find the value of k.
The actual area of a park is 15 km2, and its area
on a plan is 0.6 m2. If the scale of the plan is
1 : h, find the value of h.
Sol 5.2
1001000001 ×× =
2
1
k
4 000 000 = k2
k = 2 000
Sol )(
)()(15 ×× =
)(
)(
)(
=
5. On a floor plan of scale 1 : 120, the area of
a shop is 6 cm2. Find the actual area of the
shop in m2.
Let x m2 be the actual area of the shop.
)(
)()( ××x=
)(
)(
)(
=
6. On a map of scale 1 : 10 000, the area of
Victoria Park is 19 cm2. Find the actual
area of Victoria Park in m2.
○○○○→→→→ Ex 9E 13
area
= 162 cm2 area = 176 cm2 4 cm 5 cm
C D
2
2
1
2
1
=ℓ
ℓ
A
A
1 m = 100 cm
1 m2
= 100 × 100 cm2
1 km = 1 000 m
1 km2
= ( ) × ( ) m2
141
9.5B Similar Solids
For two similar solids,
if V1 and V2 denote their volumes, A1 and A2 denote the areas of any two corresponding
surfaces, 1ℓ and
2ℓ denote any two corresponding lengths, then: 2
2
1
2
1
=ℓ
ℓ
A
A and
3
2
1
2
1
=ℓ
ℓ
V
V
Example 3 Instant Drill 3
The figure shows two similar solids A and B.
Find the ratio of
(a) the total surface area of A to that of B,
(b) the volume of A to that of B.
The figure shows two similar solids C and D.
Find the ratio of
(a) the total surface area of C to that of D,
(b) the volume of C to that of D.
Sol (a) B
A
of area surface Total
of area surface Total =
2
3
5
=9
25
Sol (a) D
C
of area surface Total
of area surface Total =
)(
)(
)(
=
(b) B
A
of Volume
of Volume =
3
3
5
=27
125
(b) D
C
of Volume
of Volume =
)(
)(
)(
=
7. The figure shows two similar solids P and
Q.
Find the ratio of
(a) the total surface area of P to that of Q,
(b) the volume of P to that of Q.
8. The figure shows two similar solids R and
S.
Find the ratio of
(a) the curved surface area of R to that of S,
(b) the volume of R to that of S.
○○○○→→→→ Ex 9E 7, 8
9 cm 6 cm
P Q
12 cm R S
15 cm
3 cm 5 cm A B
2 m 1 m C D
142
9. The heights of the two similar cylinders as
shown below are 8 cm and 12 cm. If the
total surface area of the small cylinder is
28 cm2, find the total surface area of the
large cylinder.
10. The heights of the two similar prisms as
shown below are 21 cm and 15 cm. If the
volume of the large prism is 171.5 cm3,
find the volume of the small prism.
Let x cm2 be the total surface area of the
large cylinder.
2
2
cm )(
cm )(=
)(
)(
)(
=
11. The ratio of the heights of two similar
circular cones is 1 : 5. If the curved surface
area of the large cone is 100 cm2, find the
curved surface area of the small cone.
12. The ratio of the lengths of a pair of
corresponding sides of two similar solids
is 4 : 3. If the volume of the small solid is
189 cm3, find the volume of the large
solid.
○○○○→→→→ Ex 9E 9−12
8 cm 12 cm 15 cm 21 cm
2
2
1
2
1
=ℓ
ℓ
A
A
3
2
1
2
1
=ℓ
ℓ
V
V
143
13. In the figure, the pyramid VABCD is divided by a plane parallel to its
base into two parts VEFGH and ABCDEFGH. It is given that
BC : GH = 3 : 1.
(a) If the area of quadrilateral ABCD is 63 cm2, find the area of
quadrilateral EFGH.
(b) If the volume of VEFGH is 21 cm3, find the volume of VABCD.
○○○○→→→→ Ex 9E 14
Level Up Question
14. In the figure, there is some water in a vertical right conical vessel.
The area of the surface in contact with the water is
4
1 of that of the
curved surface inside the vessel.
(a) Find
2
1
r
r.
(b) If the volume of water in the vessel is 17 cm3, find the capacity
of the vessel.
A B
C
G
E
F
H
D
V
r2 r1
If a small pyramid is removed
by cutting a large pyramid
along a plane parallel to
its base, the part removed
is similar to the original
solid.
144
New Century Mathematics (2nd Edition) 3B
9 Area and Volume (III)
Level 1
In each of the following, figure A and figure B are similar. Find the ratio of the area of A to that of B.
[Nos. 1–2]
1. 2.
Find the unknown in each of the following pairs of similar plane figures. [Nos. 3–5]
3. 4.
5.
In Nos. 6–7, solid A and solid B are similar. Find the ratio of
(a) the curved surface area of A to the curved surface area of B,
(b) the volume of A to the volume of B.
6. 7.
A
B
3 cm
2 cm A
B
8 cm
10 cm
area = 2.7 cm2
5 cm 3 cm
area = T cm2
area = P m2
area = 324 m2
28 m
36 m
32 cm
area = 16 cm2 area = 9 cm2
x cm
7 m
3.5 m A B
15 cm
18 cm
B A
Consolidation Exercise
9E
145
scale 1 : k
Find the unknown in each of the following pairs of similar solids. [Nos. 8–11]
8. 9.
10. 11.
12. The figure shows a map of an island of scale 1 : k. The area of the
island on the map is 9 000 cm2 and the actual area of the island is
3 600 000 m2. Find the value of k.
13. In the figure, the pyramid VXYZ is divided by a plane parallel to its base into two parts VPQR
and PQRZXY. It is given that VX = 2VP.
(a) If the area of △PRQ is 85 cm2, find the area of △XZY.
(b) If the volume of pyramid VXYZ is 3 200 cm3, find the volume of pyramid VPQR.
4 cm
total surface area
= 200 cm2
total surface area
= x cm2
3 cm 15 m 9 m
volume = 24 000 m3 volume = v m3
h cm
3 cm
base area
= 36 cm2
base area
= 81 cm2
x mm
1 mm
volume = 125 mm3 volume = 64 mm3
V
P Q
R
X Y
Z
146
A
B
C
E
D
R Q
P S
Level 2
14. In the figure, ABCDE and ASRQP are two similar pentagons. If
AB : AS = 2 : 3, find the ratio of the area of pentagon ABCDE to
that of polygon BCDEPQRS.
15. In the figure, ABE and ACD are straight lines. BC // ED and AC : CD = 3 : 1. It is given that the
area of △ADE is 96 cm2. Find the area of the quadrilateral BCDE.
16. The curved surface areas of two similar cylinders P and Q are 810 cm2 and 250 cm2
respectively.
(a) Find the ratio of the base diameter of P to that of Q.
(b) A student claims that the volume of P is 6 times that of Q. Do you agree? Explain your
answer.
17. The figure shows two similar rectangular bars. The total surface area of the shorter bar is 64%
of that of the longer bar. If the volume of the shorter bar is 320 cm3, find the volume of the
longer bar.
A
B C
D E
147
18. The figure shows two dolls which are similar solids. The ratio of the volumes of the two dolls is
27 : 125.
(a) Find the height of the smaller doll : the height of the larger doll.
(b) If the area of the face on the larger doll is 150 cm2, find the area of the face on the smaller
doll.
19. In the figure, a right circular cone P is divided into a frustum X and a right circular cone Y. The
cones P and Y are similar. The heights of X and Y are 6 cm and 24 cm respectively.
(a) Find the volume of the frustum X : the volume of the cone Y.
(b) If the curved surface area of Y is 400 cm2, find the curved surface area of X.
20. The figure shows a right rectangular pyramid VTUXY. It is cut by a plane parallel to its base into
pyramid VPQRS and frustum PQRSTUXY. It is given that
area of △VPS : area of quadrilateral PSTU = 4 : 5. If the volume of pyramid VTUXY
is 1 080 cm3, find the volume of frustum PQRSTUXY.
6 cm
P
24 cm
X
Y
U
V
T
X Y
P S
Q R
148
Answer
Consolidation Exercise 9E
1. 4 : 9
2. 16 : 25
3. 7.5
4. 196
5. 24
6. (a) 4 : 1 (b) 8 : 1
7. (a) 25 : 36 (b) 125 : 216
8. 112.5
9. 5 184
10. 4.5
11. 1.25
12. 2 000
13. (a) 340 cm2 (b) 400 cm3
14. 4 : 5
15. 42 cm2
16. (a) 9 : 5 (b) no
17. 625 cm3
18. (a) 3 : 5 (b) 54 cm2
19. (a) 61 : 64 (b) 225 cm2
20. 760 cm3