9781139252386 w 01

C H A P T E R

1Length, area and volume

What you will learn1.1 Conversion of units

1.2 Length

1.3 Pythagoras’ theorem

1.4 Area

1.5 Surface area—prisms and pyramids

1.6 Surface area—spheres and cones

1.7 Volume—prisms, pyramids and cones

1.8 Volume—spheres

1.9 Scientific applications

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ISBN: 978-1-139-25238-6 Photocopying is restricted under law and this material must not be transferred to another party

© David Greenwood, David Robertson 2006 Cambridge University Press

VELSNumber

Carry out exact arithmeticcomputations involving fractions andirrational numbers such as square rootsUse appropriate estimates to evaluatethe reasonableness of the results ofcalculations involving rational andirrational numbersCarry out computations to a requiredaccuracy in terms of decimal place

SpaceRepresent two- and three-dimensionalshapes using lines, curves, polygonsand circlesMake representations usingperspective, isometric drawings, netsand computer-generated imagesRecognise and describe boundaries,surfaces and interiors of common planeand three-dimensional shapes

Measurement, chance and dataEstimate and measure length, area,surface area, mass, volume, capacityand angleSelect and use appropriate units,converting between units as requiredCalculate constant rates such as thedensity of substances, concentration offluids, average speed and pollution levelsDecide on acceptable or tolerablelevels of error in a given situationInterpret and use mensuration formulasfor calculating the perimeter, surfacearea and volume of familiar two- andthree-dimensional shapes and simplecomposites of these shapesUse Pythagoras’ theorem to obtainlengths of sides, angles and the area of right-angled triangles

Working mathematicallyFormulate generalisations andarguments in natural language and symbolic formChoose, use and develop mathematicalmodels and procedures to solveproblems set in a wide range ofpractical contextsUse technology to analyse theexponential function

Tower of TerrorThe Tower of Terror at Dreamworld onthe Gold Coast is one of the fastestrides in the world. The tower’selectromagnetically powered ‘Escapepod’ accelerates to a speed of160 km/h in 7 seconds. After rising toa height of 38 stories, there is a 100 mfreefall where riders experience 6.5seconds of weightlessness.

After just 18 months of operation,the Tower of Terror introduced its 1 000 000th passenger and hadtravelled 56 668 km—the equivalentdistance to almost 1.5 trips around theworld. Over 600 tonnes of steel and1200 cubic metres of concrete wereused to build the Tower of Terror.

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Essential Mathematics VELS Edition Year 10

Do now

4

Skillsheet

T EACHE R

1 Evaluate the following.

a b cd e f

2 Evaluate the following.

a if and b if and c if and d if and

3 Find the perimeter of these shapes.

a b c

4 Use these rules to find the circumference (C ) and area (A) of these circles correct to onedecimal place. Remember and where r is the radius.

a b c

5 Use Pythagoras’ theorem to find the value of the unknown in thesetriangles. Round to one decimal place where necessary.

a b c

Answers1 a 200 b 0.05 c 2.3 d 430 e 62.9 f 1 38 0000 2 a 21 b 30 c 3.8 d 10 3 a 12 cm b 7 m c 32 cm4 a b c5 a b c a � 5c � 2.2c � 5

A � 47.8 km2C � 24.5 km,A � 113.1 cm2C � 37.7 cm,A � 12.6 m2C � 12.6 m,

12

a13

2

1 c

4

3c

a2 � b2 � c2

3.9 km12 cm2 m

A � �r2C � 2�r

5 cm

3 cm

3 m

1 m

3 cm

h � 4b � 3a � 2,A � 12(a � b)h

h � 3.8b � 2A � 12b � h

h � 3b � 10A � b � hw � 7l � 3A � l � w

1.38 � (1000)262900 � 10000.043 � (100)2

230 � (10)25 � (10)22 � 100

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Chapter 1 — Length, area and volume

The most commonly used metric units are based on millimetres (mm), centimetres (cm),metres (m) and kilometres (km).

Simple one, two or three-dimensional diagrams can help to convert units of length, area andvolume. When converting between square metres and square centimetres forexample, it helps to draw a square of one of the larger unit, labelling the side lengths usingthe smaller unit. The multiplying factor can then be determined as shown.

Many non-metric imperial units of measurement are commonly referred to in our society.Some of these and other metric units are described here.

LENGTH InchesFeetMiles

AREA SquaresHectares (ha)Acre

VOLUME Millilitres (mL)Litres (L) 1 litre � 1000 mL � 1000 cm3

1 millilitre � 1 cm2

1 acre � 1640 square miles

1 hectare � 10 000 m2

1 square � 100 square feet � 9.29 m2

1 mile � 1.609 km � 1609 m1 foot � 12 inches � 30.48 cm1 inch � 2.54 cm � 25.4 mm

100 cm

100 cm1 m2 1 m2 = 100 × 100 = 10000 cm2

(cm2)(m2)

5

1.1 Conversion of units

Key ideas

When converting units:• use a diagram to help determine the multiplying or dividing factor• use multiplication if converting to a smaller unit• use division if converting to a larger unit

Base unit Length Area Volume

mm

10 (10)2 (10)3

(100)3

(1000)3

(100)2

(1000)2

100

1000

cm

km

m××÷

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Example 1

Convert these measurements into the units given in brackets.

a 8.2 km (m) b c 3.72 cm3 (mm3)930 cm2 (m2)

ExplanationSolution

a

b

c

� 3720 mm3

� 3.72 cm3 � 3.72 � 1000� 1000 mm3

1 cm3 � (10)3 mm3

� 0.093 m2

� 930 cm2 � 930 � 10 000� 10 000 cm2

1 m2 � (100)2 cm2

� 8200 m� 8.2 km � 8.2 � 1000

1 km � 1000 m

10 mm

10 mm

10 mm

1 cm3

100 cm

100 cm

1 m2

1 km

1000 m

1AExercise

1 Convert the following length measurements into the units given in brackets.

a 4 cm (mm) b 0.096 m (cm) c 0.001 km (m)d 300 m (km) e 800 cm (m) f 297 m (km)g 0.0127 m (cm) h 5102 mm (cm) i 0.0032 km (m)

2 Convert the following area measurements into the units given in brackets.

a b c

d e fg h i

3 Convert these volume measurements into the units given in brackets.

a b cd e fg h i

4 Find the total sum of these measurements. Express your answer in the units given inbrackets.

a 10 cm, 18 mm (mm) b 1.2 m, 19 cm, 83 mm (cm)c 453 km, 258 m (km) de fg hi 6 640 000 mm3, 0.000 32 m3 (cm3)

0.000 51 km3, 27 300 m3 (m3)482 000 mm3, 2.5 cm3 (mm3)0.000 03 km2, 9 m2, 37 000 000 cm2 (m2)0.3 m2, 251 cm2 (cm2)

11.5 cm2 (cm2)400 mm2,

2.094 cm3 (mm3)0.000 001 km3 (m3)0.13 m3 (cm3)762 000 cm3 (m3)28 300 000 m3 (km3)5700 mm3 (cm3)0.015 km3 (m3)0.2 m3 (cm3)2 cm3 (mm3)

0.0027 km2 (m2)537 cm2 (mm2)0.023 m2 (cm2)0.0001 km2 (m2)5 km2 (m2)0.5 m2 (cm2)

205 000 m2 (km2)29 800 cm2 (m2)3000 mm2 (cm2)

1aExample

1bExample

1cExample

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5 Convert these special measurements into the units given in brackets. Hint: Use theconversion information given on page 5 to help you.

a 5.5 miles (km) b 54 inches (feet) c 10.5 inches (cm)d 3218 m (miles) e fg 8.2 L (mL) h i 10 squares (square feet)j k l

6 An athlete has completed three quarters of a 5-km marathon. How many metres doesthe athlete have left to run?

7 A snail is moving at a rate of 43 mm every minute. How many centimetres will thesnail move in 5 minutes?

8 A rectangle of length 30 cm and width 185 mm is to be cut from a piece of papermeasuring in area. Find the area remaining in

9 These measurement conversions require more that one step as they skip one metricunit. Convert to the measurement shown in the brackets.

a 20 000 cm (km) b 0.0045 m (mm) cd e f

10 A farmer intends to plough an area of forvegetable growing and 1.5 ha for cropping. Find thetotal area, in hectares, that the farmer will plough.

11 How many containers holding (1 L) ofwater will you need to fill

12 A crate with a volume of is packed with1585 spheres each of volume Find thevolume in of air space remaining in the crate.m3

172 cm3.0.5 m3

1 m3?1000 cm3

3000 m2

0.000 000 0002 km3 (cm3)670 000 mm3 (m3)0.000 22 km2 (cm2)10 m2 (mm2)

cm2.0.5 m2

152 000 mL (m3)1 km2 (ha)2 m3 (L)5.5 m3 (mL)

247 cm3 (L)5.7 ha (m2)

7

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Enrichment

13 While measurement in the modern world is dominated by metric units, it can be usefulto understand some commonly used non-metric units. Round each of the following toone decimal place and use the information given in the introduction to help.

a Find the area in of 30 squares of housing space.b Find the number of ha in 100 square miles of bush land.c Find the area in of a -acre block.d Find the area in ha of 1000 acres of land.

14 As a backyard landscaper, Larry requires enough soil mulch to cover an area equal to of an acre. Other information regarding the thickness of mulch for particular areas for the landscaping job is as follows:

• 100 square metres requires mulch 10 cm thick• 850 square feet requires mulch 20 cm thick• ha requires mulch 25 cm thick• the remainder requires mulch 5 cm thick

Find how many cubic metres of mulch are required for the complete job.

150

18

14m2

m2

Th




Length measurements are common in many areas of mathematics, science and engineeringand are clearly associated with the basic measures of perimeter and circumference, whichwill be studied here.

1.2 Length

8

Key ideas

Perimeter is the distance around the outside of a 2-dimensional shape.

Line segments with the same markings are of equal length.

Circumference is defined as the distance around the outside of a circle.

Circumference � 2�r or �d

z y

x

rd

P � 2x � y � z

Example 2

ExplanationSolution

Consider the given 2-dimensional shape.

a Find the perimeter of the shape if b Find x if the perimeter is 11.9 m.c Write an expression for x in terms of the

perimeter P.

x � 2.6.

a

b

c

� x � P � 10� 10 � x

P � 4.5 � 2.1 � 3.4 � x

� x � 1.9� 10 � x

11.9 � 4.5 � 2.1 � 3.4 � x

� 12.6 m Perimeter � 4.5 � 2.1 � 3.4 � 2.6 Simply add up the lengths of all sides

using

UseSimplifySubtract 10 from both sides

Use P for the perimeterSimplifyMake x the subject

P � 11.9

x � 2.6

4.5 m

3.4 m

x m2.1 m

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Chapter 1 — Length, area and volume 9

Example 3

ExplanationSolution

If a circle has radius r cm, find the following and round to two decimal places, where necessary.

a The circumference of a circle if b A rule for r in terms of the circumference Cc The radius of a circle with a circumference of 10 cm

r � 2.5

a

b

c

� 1.59

�102�

r �C

2�

� r �C

2�

C � 2�r

� 15.71 cm� 2�(2.5)

Circumference � 2�r Write the ruleSubstituteEvaluate and round

Write the rule

Divide both sides by

SubstituteEvaluate and round

C � 10

2�

r � 2.5

r cm

1BExercise

1 Consider the given 2-dimensional shape.

a Find the perimeter of the shape if b Find x if the perimeter is 23 cm.c Write an expression for x in terms of the perimeter P.

2 Consider the given 2-dimensional shape.

a Find the perimeter of the shape if b Find x if the perimeter is 12 m.c Write an expression for x in terms of the

perimeter P.

3 Find the perimeter of these shapes.

a b c

32 cm

12 cm

34 cm

0.5 km

2.6 km

185 m

220 m

x � 2.3.

x � 7.11 cm

x cm

6 cm

3.1 m

4.3 mx m

2Example

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4 If a circle has radius r cm, find the following and round to two decimal places, where necessary.

a The circumference of a circle if b A rule for r in terms of the circumference Cc The radius of a circle with a circumference of 35 m

5 Find the circumference of these circles, correct to two decimal places.

a b c

6 a Rearrange the formula for the circumference of a circle, to express r interms of C.

b Find, correct to two decimal places, the radius of the circle with the givencircumferences.i 35 cm ii 1.85 m iii 0.27 km

7 Find expressions for the perimeter (P) of these shapes using the pronumerals given.

a b c

d e f

8 Find the value of x for these shapes with the given perimeters.

a b c

Perimeter = 0.072

2x

x

5.3Perimeter = 22.9

x

2

5

7

Perimeter = 17

sla

b

x

z

y

l

w

l

C � 2�r,

1.07 km

19.44 mm18 cm

r � 12

r m

3Example

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d e f

9 Find the perimeter of these composite shapes correct to two decimal places.

a b c

d e f

126

3.6

2.210

0.3

7.9

1.5

3.6

Perimeter = 10.5

2x

1.5x

x

Perimeter = 46.44

3.72

11.61

3x7.89

Perimeter = 16.2

2.8 2x

1111

Enrichment

10 Consider a rectangle with perimeter P, length l and width w.

a Express l in terms of w and P.b Express l in terms of w ifc If state the range of all possible values of w.d If state the range of all possible values of l.e Explore other common shapes and answer similar question to parts a to d

above. You may need to choose new pronumerals.

11 The exact perimeter of a circle with radius 2 can be expressed as Find the perimeter of these shapes using exact values.

a b c

12 Use exact values (e.g. to find the answers to each shape in question 9 above.6�)

1 km

5 m3 cm

C � 2�(2) � 4�.

P � 10,P � 10,

P � 10.

l

w

Th

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Pythagoras’ theorem is a relationship linking the side lengthsof a right-angled triangle. Given two sides of a right-angledtriangle, Pythagoras’ theorem can be used to find the lengthof the third side. This has applications in both two and threedimensions.

1.3 Pythagoras’ theorem

12

Key idea

Pythagoras’ theorem states that:The sum of the squares of the two shorter sidesof a right-angled triangle equals the square ofthe hypotenuse.

i.e. in this triangle a2 � b2 � c2

ca

b

Example 4

ExplanationSolution

Find the length of the unknown side in these right-angled triangles, correct to two decimal places.

a b 1.1 m

1.5 my m

x cm

9 cm

5 cm

a

The length of the unknown side is 10.30 cm.

b

The length of the unknown side is 1.02 m.� 1.02

� y � 11.04� 1.04

y2 � 1.52 � 1.12

y2 � 1.12 � 1.52

a2 � b2 � c2

� 10.30� x � 1106

� 106� x2 � 52 � 92

c2 � a2 � b2

Substitute the two shorter sides and or and Square root both sides

Substitute the shorter side and the hypotenuse Subtract from both sidesSquare root both sides

1.12

c � 1.5b � 1.1

b � 5a � 9b � 9a � 5

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Example 5

ExplanationSolution

Consider a cuboid ABCDEFGH with the side lengths and Find:

a BE, leaving your answer in surd formb BH, correct to two decimal places

EH � 2.AE � 4AB � 7,

a

b

� 8.31� BE � 169

� 69� 4 � 65

� BH2 � 22 � (165)2

c2 � a2 � b2

� BE � 165� 65

� BE2 � 42 � 72

c2 � a2 � b2

Draw the appropriate triangle

Substitute a � 4 and b � 7Solve for BE exactlyLeaving intermediate answers in surd formreduces the chance of accumulating errors.

Draw the appropriate triangleSubstitute HE � 2 and

Note (165)2 � 165 � 165 � 65

EB � 165

H

D

A

E

B

C

G

F

1CExercise

1 Use Pythagoras’ theorem to find the length of the hypotenuse for these right-angledtriangles. Round your answers to two decimal places where necessary.

a b c

d e f

72.1 cm

27.3 cm

0.37 km 0.21 km0.2 mm

1.8 mm

15 km

7 km10 m

5 m3 cm

4 cm

4aExample

E

AB

4

7

H

2

EB

√65

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2 Find the length of the unknown side in these right-angled triangles, correct to twodecimal places.

a b c

d e f

3 Use Pythagoras’ theorem to find the distance between points A and B in thesediagrams, correct to two decimal places.

a b c

d e f

4 For each of the cuboids ABCDEFGH, find:

i BE, leaving your answer in surd formii BH, correct to two decimal placesa b c

d H G

F

9

82

BA

D

E

C

G

CD

H

B

EF

A

7

3

1

H D

GC

BF

EA

14

8

4

H G

F

C

E

53

2

B

D

A

BA

19.7 km17.2 km

14.3 km

AB 0.5 km

1.8 km2.1 km5.3 cm

5.3 cm

A

B

BA

49 m

26 m

35 m

A

B

1.6 cm

3.5 cm

3 cm

1.9 cm

1.2 m

2.6 m

B

A

0.14 cm

0.11 cm24.2 cm

19.3 cm0.71 cm

1.32 cm

0.7 m

0.3 m

12 m

9 m

5 m2 m

14

4bExample

5Example

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5 Find the value of x, correct to two decimal places, in these 3-dimensional diagrams.

a b c .

d e f

6 Starting at a point A on a runway, an aircraft increases its speed and, after 400 metres, lifts off at B. At a height of 350 metres the aircraft has travelled a further 750 metres horizontally. At this point, find the total actual distance that the aircraft has travelled, correct to the nearest metre.

7 a Find the length of the longest rod, correct to one decimal place, that will fit insidethese objects:i cylinder with diameter 10 cm and height 20 cmii cuboid with side lengths 10 cm, 20 cm and 10 cm

b Investigate the length of the longest rod that will fit in other solids.

x cm

4.04 cm

2.93 cm

5.31 cm

6.7 km

6.2 km

8.2 km

x km

1.5 m

x m

2 m

3 m

3 cm

5 cmx cm

x m

15.5 m

3.7 mx mm

9.3 mm

11.4 mm

15

350 m

A B

Enrichment

8 A piece of ribbon is used to decorate the interior of a room with dimensions 4.5 metres long, 3.5 metres wide and 3 metres high, as shown.

a Find the length of ribbon, correct to two decimal places, required to connect from:i A to Hii E to Biii A to Civ A to G via Cv E to C via Dvi E to C directly

b Find the shortest length of ribbon required, correct to two decimal places, toreach from A to G if the ribbon is not allowed to reach across open space.

3 m

4.5 m

3.5 m

A B

CD

FE

H G

Th

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Area is a measure of surface and is expressed as a number ofsquare units.

Clearly, by the inspection of a simple diagram, a rectangle withside lengths 2 m and 3 m has an area of 6 square metres or

For rectangles and other basic shapes, we can use area formulas tohelp us count the number of square units.

6 m2.

1.4 Area

16

3 m

2 m

Area = 6 m2

Key ideas

The area of a 2-dimensional shape can be defined as the number of square unitscontained within its boundaries. Some common area formulas are as follows:

The rule for the area of a circle is:

Area � �r 2

Trapeziuma

h

b

Parallelogram

h

b

Rhombus

xy

Triangle

h

b

Rectangle

w

l

Square

2

l

r

Example 6

Find the area of these basic shapes, correct to two decimal places where necessary.

a b c

1.06 km3.3 m

5.8 m

3 cm

2 cm5 cm

Area � l2

Area � xy12

Area � bh

Area � (a � b)h12

Area � lw Area � bh12

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ExplanationSolution

a

b

c

� 0.88 km2

� �(0.53)2

A � �r2

� 9.57 m2

�12

(5.8)(3.3)

A �12

bh

� 8 cm2

�12

(3 � 5)2

A �12

(a � b)h The shape is a trapezium

Substitute and

Simplify and include the correct units

The shape is a triangle

Substitute and

Simplify and include the correct units

The shape is a circleThe radius is half the diameterRound to the correct number of decimal places

h � 3.3b � 5.8

h � 2b � 5a � 3,

Example 7

Find the value of the pronumeral for these basic shapes, rounding to two decimal placeswhere necessary.

a b 1.3 m

a m0.4 m

Area = 0.5 m2

2.3 mm

Area = 11 mm2

l mm

ExplanationSolution

a

b

� a � 1.2 2.5 � a � 1.3 0.5 � 0.2(a � 1.3)

0.5 �12

(a � 1.3) � 0.4

A �12

(a � b)h

� 4.78

� l �112.3

11 � l � 2.3A � lw Use the rectangle area formula

Substitute and

Divide both sides by 2.3

Use the trapezium area formula

Substitute and SimplifyDivide both sides by 0.2Subtract 1.3 from both sides

h � 0.4b � 1.3A � 0.5,

w � 2.3A � 11

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1DExercise

1 Find the area of these basic shapes, rounding to two decimal places where necessary.

a b c

d e f

g h i

j k l

2 Find the value of the pronumeral for these basic shapes, rounding to two decimalplaces where necessary.

a b c

d e f

g h i

Area = 26 km2

d m

Area = 0.21 m2

d mr cm

Area = 5 cm2

18 m

x mArea = 80 m2

Area = 10.21 mm2

5.34 mm

h mm2.8 m

1.3 m

Area = 2.5 m2

a m

Area = 1.3 km2

1.8 km

h km

Area = 206 m2

l m

5.2 cm

Area = 15 cm2

w cm

√10 cm

√6 cm√2 cm

0.82 m

1.37 m

√3 km

km

km

23

14

0.4 mm

0.25 mm

64 m23 m

20 cm10 cm

7 m

0.3 mm

0.1 mm

0.2 mm

1.3 km

2.8 km

10.5 m

5.2 m

5 cm

6Example

7Example

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3 A lawn area is made up of a semicircular region with diameter 6.5 metres and a triangular region of length 8.2 metres, as shown. Find the total areaof lawn, to one decimal place.

4 An L-shaped concrete slab being prepared for the foundation of a new house is made up of two rectangles with dimensions 3 metres by 2 metres and 10 metres by 6 metres.

a Find the total area of the concrete slab.b If two bags of cement are required for every 5 square metres of concrete, how

many whole bags of cement will need to be purchased for the job?

5 Find the area of these composite shapes rounding to two decimal places wherenecessary.

a b c

d e f0.3 m

0.7 m

0.25 m

28 km

26 km

18 km

4.2 mm

1.8 m

1.7 m

1.6 m

4 m

7 m

10 m5 cm


Enrichment

6 Consider a trapezium with area A, parallel side lengths a and b and height h.

a Rearrange the area formula to express a in terms of A, b and h.b Hence, find the value of a for these given values of A, b and h.

iiiiii

c Sketch the trapezium with the dimensions found in part b (iii) above. What shapehave you drawn?

7 Use exact values (e.g. to find the area of the shapes in question 5 a, c and e.1 � 3�)

A � 10, b � 5, h � 4A � 0.6, b � 1.3, h � 0.2A � 10, b � 10, h � 1.5

A � 12(a � b)h

a

b

h

6.5 m

8.2 m

Th

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Three-dimensional objects have outside surfaces which can becategorised into one or more different shapes.

A cylindrical can, for example, has two circular ends and a curved surface which could be rolled out to form a rectangle.

1.5 Surface area—prisms and pyramids


Key ideas

The total surface area (TSA) of a three-dimensional object can be found by finding thesum of the areas of each of the shapes that make up the surface of the object.

A net is two-dimensional illustration of all the surfaces of a solid object. The following isthe net for a cyliner.

Diagram Net

TSA � 2 circles � 1 rectangle� 2�r 2 � 2�rh� 2�r (r � h)

h

r

Example 8

Find the total surface area (TSA) of these objects. Round to two decimal places wherenecessary.

a b c

22 mm

25 mm1.7 m

5.3 m8 cm

5 cm

3 cm

2p r

r

r

h

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ExplanationSolution

a

b

c

� 1725 mm2

� 252 � 4 �12

� 25 � 22

� l2 � 4 �12

bh

TSA � 1 square � 4 triangles

� 74.77 m2

� 2�(1.7)2 � 2�(1.7) � 5.3� 2�r2 � 2�rh

TSA � 2 circles � 1 rectangle

� 158 cm2

TSA � 2 � (8 � 3) � 2 � (5 � 3) � 2 � (8 � 5) Draw the net of the solid

5 cm

3 cm

3 cm3 cm

8 cm

5 cm

5.3 m

1.7 m

1.7 m

2πr

25 mm

22 mm

1EExercise

1 Find the TSA of these objects. Round your answers to two decimal places wherenecessary.

a b c

d e f 0.2 m

0.3 m

5 cm

8 cm

12.8 m

9.2 m

26 cm

11 cm

2.7 mm

1.3 mm5.1 mm

3 cm

3 cm6 cm

8Example

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2 Find the TSA of these objects.

a b c

d e f

3 Use Pythagoras’ theorem to determine any unknown side lengths and find the TSA ofthese objects, correct to one decimal place.

a b c

d e f

4 Find the TSA of the outside of apipe of radius 85 cm and length4.5 m correct to two decimalplaces.

5 What is the minimum amount ofpaper required to wrap a box withdimensions 25 cm wide, 32 cmlong and 20 cm high?

5.21.8

26

3320

12

7

810

1.8 3.2

1.4

2

3

7

1.4 m

2.1 m

4.8 m

2.3 m

1.9 m

0.5 cm 1.2 cm

1.5 cm

0.76 cm 0.8 cm

8.5 mm 10 mm

11 mm3 mm

25 m

8.66 m

10 m5 mm

7 mm

4 mm

3 mm1.2 cm

22

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Enrichment

7 Find the exact TSA in terms of for a cylinder with the given dimensions.

a and

b and

8 Prove the given TSA rules by drawing a net and showing working.

a b c

9 If the TSA of a cylinder is given by the rule find the height, totwo decimal places, of a cylinder if the radius is 2 m and the TSA is:

a b

10 Can you find the exact radius of the base of a cylinder if its TSA is and itsheight is 3 cm?

11 For a cylinder of fixed TSA, try to find the maximum and minimum radius and heightlengths possible. Explore different TSA measurements and write a conclusion.

cm28�

122 m235 m2

TSA � 2�r(r � h),

h

b

TSA = b(2h + b)

l

TSA = 6l2

h

r

TSA = 2πr(r + h)

h � 5r � 12

h � 2r � 1

�

6 The roof of a rectangular dog kennel is to be madewith sheets of tin cut into two isosceles trianglesand two isosceles trapeziums as shown.

a Use Pythagoras’ theorem to find:i the height of the isosceles trapeziums

correct to 3 decimal placesii the height of the isosceles triangles

correct to 3 decimal placesb Find the TSA of tin required to make the roof of the kennel, correct to one

decimal place.

(h2)

(h1)

50 cm

h1

h2

90 cm

80 cm

1 m

60 m

Th

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The rule for the surface area of a cone can beconsidered by drawing a net including a circle(base) and sector (curved surface). The surfacearea formula of a sphere is more difficult toprove and can be achieved using mathematicsnormally studied at Year 12 or university level.

1.6 Surface area—spheres and cones

24

Key ideas

Cone total surface area

Sphere total surface area

TSA (sphere) � 4�r 2

r

� TSA (cone) � �r 2 � �rs � �r (r � s)

Example 9

Find the total surface area (TSA) of these solids, correct to two decimal places.

a sphere with radius 5 mm

b cone with radius 2 m and slant height 4.5 m

4.5 m

2 m

5 mm

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ExplanationSolution

a

b

� 40.84 m2

� �(2)2 � �(2) � (4.5) TSA � �r2 � �rs

� 314.16 mm2

� 4�(5)2

TSA � 4�r2 Write the appropriate formulaSubstituteEvaluate, round and use the correct units

Write the appropriate formulaSubstitute and Evaluate, round and use the correct units

s � 4.5r � 2

r � 5

1FExercise

1 Find the TSA of a sphere with the given dimensions, correct to two decimal places.

a radius 2 cm b radius 5 m c radius 10.5 md radius e radius f diameter � km

2 Find the TSA of these cones, correct to two decimal places.

a b c

3 Find the area of the curved surface only of these cones, correct to two decimalplaces.

a b c

4 Two marbles have radii 1 cm and 2 cm. Find thedifference in their total surface areas, correct to onedecimal place.

5 Party hats A and B are in the shape of open cones withno base. Hat A has radius 7 cm and slant height 25 cmand hat B has radius 9 cm and slant height 22 cm.Which hat has a greater surface area?

6 A cone has height 10 and radius 3.

a Use Pythagoras’ theorem to find the slant height of the cone,and express your answer using surds.

b Find the TSA of the cone, correct to one decimal place.

48 mm

26 mm

1.1 cm

1.5 cm

10 m

2 m

15 km

11 km0.5 m

0.8 m12 mm

9 mm

17 cm13 mm

10

3

9aExample

9bExample

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7 A hemisphere sits on a cone and two height measurementsare given as shown. Find:

a the radius of the hemisphereb the slant height of the cone, correct to two decimal placesc the TSA of the object, correct to one decimal place

8 Find the TSA for these objects, correct to two decimal places.

a b c

d e f3 cm

2 cm

√2 cm2 m

5 m

1 m3 m

8 mm

4 mm

0.7 cm

1.3 cm

5 m

26

15 cm

10 cm

Enrichment

9 Using the formula find the slant height (s) of a cone to onedecimal place with radius 2 cm and TSA as follows.

a b

10 Consider the rule for the surface area S of a sphere with radius r:

a Express r in terms of S.b Find the radius of a sphere with these surface areas, correct to two decimal places.

i ii

11 A sculpture consists of a square-basedpyramid sitting on the flat side of ahemisphere of radius 5 metres, as shown.The total height of the sculpture is 11 metres.

a Find the side length of the base of thepyramid in surd form.

b Find the height of the pyramid.c Find the height of the triangular faces

on the pyramid in surd form.d Find the TSA of the sculpture to two decimal places.

12 Express the TSA of the objects in question 8 using exact values.

13 By researching the radius of the earth, find an approximate surface area of the earthin km2.

0.35 m2100 cm2

S � 4�r2.

150 cm220 cm2

TSA � �r(r � s),

5 m

11 m

Object

5 mPlan view

Th

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Volume is the word used to describe theamount of space contained within the outsidesurfaces of a three-dimensional object, and itis measured in cubic units.

Three common groups of objects consideredhere are the prism, the pyramid and the cone.A prism has a constant cross-section.

27

1.7 Volume—prisms, pyramids and cones

Key ideas

PrismsThe volume of a prism is equal to the area of the constant cross-section (A) multiplied by its height.

volume � area (cross-section) � height

For a cuboid (rectangular prism):

For a cylinder:

�r2h

Pyramids and conesThe volume of a pyramid or cone is one-third of the volume of a prism with the same base area:

For a cone:

�r2hvolume �13

�13

Ah

volume �13

area (base) � height

volume �

volume � l � w � h

height

cross-section

lw

h

cuboid

r

h

cylinder

h

base

square-based pyramid

h

base

cone

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Example 10

ExplanationSolution

Find the volume of the following prisms, correct to two decimal places where necessary.

a b 2 cm

6 cm

4 m

3 m2 m

a

b

� 75.40 cm3

� �(2)2 � 6V � �r2h

� 24 m3

� 2 � 3 � 4V � lwh The prism is a cuboid (rectangular prism)

Substitute and

The prism is a cylinder with base area Substitute and Evaulate and round

h � 6r � 2�r2

h � 4w � 3l � 2,

Example 11

ExplanationSolution

Find the volume of this pyramid and cone, correct to two decimal places.

a b

29 mm

23 mm

1.4 m

1.2 m

1.3 m

a

b

� 4016.26 mm3

�13

�(11.5)2 � 29

�13

�r2h

V �13

Ah

� 0.73 m3

�13

(1.4 � 1.2)1.3

�13

(l � w)h

V �13

Ah

The pyramid has a rectangular base of area

Substitute and

Evaulate and round

The cone has a circular base of area

Substitute and

Evaulate and round

h � 29r � 232 � 11.5

�r2

h � 1.3w � 1.2l � 1.4,l � w

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1GExercise

10aExample

10bExample

11aExample

11bExample

1 Find the volume of the following cuboids.

a b c

2 Find the volume of the following cylinders, correct to two decimal places.

a b c

3 Find the volume of the following prisms, rounding your answers to two decimal placeswhere necessary.

a b c

d e f

4 Find the volume of the following pyramids.

a b c

5 Find the volume of the following cones, correct to two decimal places.

a b c

60 m

20 m

3.5 mm

1.6 mm

1.1 m

2.6 m

5 km

12 km

5 km

18 m

11 m

13 m

2 cm

3 cm

21.2 cm

18.3 cm

24.5 cm0.12 m

0.38 m

A = 5 cm2

A cm2 2 cm

3 m

3 m7 m

2 m

7 cm

8 cm

3.5 cm

2 km

3 km

10 km

12 m

14 m

2 cm

1.5 cm10 m

5 m

7 mm

3.5 mm10.6 mm

35 m

10 m

30 m

2 cm

4 cm5 cm

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Enrichment

12 A heap of grain in the shape of a cone is being formed when wheat is unloaded froma ship onto a dock platform. Round the following to two decimal places.

a At 1 p.m. the heap is 2 metres high and 3 metres wide.Find the volume of wheat in the heap.

b At 3 p.m. the heap is 7 metres high and 10 metreswide. Find the difference in the volume of wheat tothat at 1 p.m.

c If at 4 p.m. the heap is 12 metres wide and has a volumeof find the height of the heap.

d If at 5 p.m. the height of the heap is 10 metres and ithas a volume of find the diameter of the heap.

13 If the volume of a cylinder has the same numerical value asits TSA, express:

a r in terms of h b h in terms of r

Explore this same question for other solids.

400 m3,

350 m3,


6 What volume of ice cream, correct to one decimal place, fills a cone of radius 3 cmand height 15 cm? Assume that the ice cream is level with the top of the cone.

7 What volume of air is contained in a perspex square-based pyramid with base sidelength 8 cm and height 6 cm?

8 Find the volume of these composite objects, rounding to two decimal places wherenecessary.

a b c

d e f

9 Use the rule to find the height of a cylinder, to one decimal place, with

radius 6 cm and volume

10 Use the rule to find the base radius of a cone, to one decimal place, with

height 23 cm and volume

11 of water is poured into a rectangular tank 2 metres wide, 3 metres long and 2 metreshigh. How high up the sides of the tank, correct to one decimal place, will the water rise?10 m3

336 cm3.

V � 13�r2h

62 cm3.V � �r2h

7.1 m

10 m

2.6 m

2.8 m

4.2 m3.8

mm

1.5 mm

5.2 mm

2.5 m

15 m

7.5 m6 cm

5 cm

3 m8 m

6 m

Th

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Graphics and CAS calculator programs can be used to find various lengths, areas andvolumes for two- and three-dimensional objects.

Example: Write and execute a program which finds:a the circumference and area of a circleb the volume of a cone correct to two decimal places

Using technology to find lengths, areas and volumes

TI 84 plus family TI 89 family

a Press PRGM andselect NEW,Create New.Then type a namee.g. CIRCLE andpress b.

Type the programusing the PRGMmenus for most ofthe functions.

Use I/O for Inputand Disp

a Go to theProgram editor andselect NEW. Thentype a name e.g.circ and pressb twice.

Type the programusing the F menusfor most of thefunctions.

Use I/O for Inputand Disp

ExerciseWrite and execute programs to find the following.

a Area of a triangle. Test:i ii

b Area of a trapezium. Test:i ii

c Volume and surface area of a sphere. Test:i ii

d Volume and surface area of a cone. Test:i ii

Suggestion: Include in your program a line that calculates the slant height s.

h � 0.75 mmr � 0.5 mm,h � 15 cmr � 10 cm,

r � 0.04 mmr � 5.5 mm

h � 4.8 mb � 3.7 cm,a � 2.1 cm,h � 1 cmb � 3 cm,a � 5 cm,

h � 22 mb � 15 m,h � 1 cmb � 3 cm,

Execute/run the program by pressing PRGMand EXEC.

b Define a newprogram and type thecode.Fix 2 rounds theanswer to twodecimal places.

Execute/run the program type circ() in thehome screen and press b.

b Define a newprogram and typethe code.Fix 2 rounds theanswer to twodecimal places.

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Spheres have a circular cross-section which varies in size depending onhow far the cross-section is drawn from the centre of the sphere.

1.8 Volume—spheres

32

r

Key idea

The volume of a sphere depends on its radius r and is given by:

volume �43

�r 3

r

Example 12

ExplanationSolution

Find the volume of a sphere of radius 3 cm, correct to twodecimal places.

� 113.10 cm3�43

�(3)3V �43

�r3 Write the ruleSubstituteEvaluate and round

r � 3

3 cm

Example 13

Find the radius of a sphere with volume correct totwo decimal places.

10 m3

r

V = 10 m3

ExplanationSolution

� r � A3 15

2�� 1.34 m

152�

� r3

30 � 4�r3

10 �43

�r3

V �43

�r3

Substitute

Multiply both sides by 3Divide both sides by

Take the cube root of both sides to maker the subject

4�

V � 10

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Example 14

ExplanationSolution

Find the volume of this composite object, using exactvalues.

Radius

� 12� cm3�36�

3�

16�

3�

20�

3

�12

�43

�(2)3 �13

�(2)2(5)

V �12

�43

�r3 �13

�r2h

r � 7 � 5 � 2 cm First find the radius r cm

Substitute and

Simplify and add the fractionsExpress the exact answer in terms of �

h � 5r � 3

Volume �12

(sphere) � cone

7 cm5 cm

1HExercise

1 Find the volume of the following spheres, correct to two decimal places.

a b c

d e f

2 Find the approximate volume of air in a hot-air balloon of radius 7.5 metre, correct tothe nearest cubic metre.

3 A square box with dimensions 30 cm long, 30 cm wide and 30 cm high holds 50 tennisballs of radius 3 cm. Find:

a the volume of 1 tennis ball, correct to two decimal placesb the volume of 50 tennis balls, correct to one decimal placec the volume in the box not taken up by the tennis balls, correct to one decimal place

1.36 m0.92 km

18 cm

38 mm

0.5 m

2 cm

12Example

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4 Find the radius of these spheres with the given volumes, correct to two decimal places.

a b c

5 of water is pumped into an expanding spherical storage bag. Find the diameterof the bag after all the water has been pumped in, correct to one decimal place.

6 Find the volume of the following composite objects, using exact values.

a b c

d e f

1 cm

hollow

2 m

2.0 m

2 m

28 cm

20 cm

2 cm

13 m

19 m

1 m

3 m

800 cm3

V = 0.52 km3

r

V = 180 cm3

r

V = 15 cm3

r

34

13Example

14Example

Enrichment

7 A spherical party balloon is blown up to help decorate a room.

a Find the volume of air, correct to two decimal places, needed for the balloon to be:i 10 cm wide ii 20 cm wide iii 30 cm wide

b If the balloon pops when the volume of air reaches find the diameter of the balloon at the point when it pops, correct to two decimal places.

8 An open rectangular container 85 cm long, 30 cm wide and 45 cm high is initially fullof liquid. When 25 marbles of radius 1 cm are placed into the container, some liquidspills over the edges as a result. Find:

a the initial volume of liquid in the containerb the volume of 1 marble correct to two decimal placesc the volume of liquid that spills over the edges of the

container after all the marbles have been added, correctto one decimal place

120 000 cm3,

Th

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1.9 Scientific applications

35

Measurement calculations are commonplace inall areas of science. In this section we look attwo of these measurements: density andconcentration.

Key ideas

Density is defined as the mass or weight of a substance per cubic unit of volume.

or

Concentration is associated with the purity of dissolved substances and will be consideredhere using percentages.

concentration (%) �volume of substance

total volume�

1001

mass � density � volumedensity �mass

volume

Example 15

The density of a solid steel object is If the object iscylindrical in shape with radius 0.5 metres and height 1.5 m, findthe mass of the object, correct to one decimal place.

500 kg per m3.1.5 m

0.5 m

ExplanationSolution

The mass of the object is 589.0 kg.� 589.0

� mass � 1.178 � 500

� 1.178 m3

� �(0.5)2 � 1.5V � �r2h First find the volume of the the cylinder

Use a sufficient number of decimalplaces to ensure you do not accumulateerrors later

mass � density � volume

Example 16

An experiment involves mixing of pure acid into a container half full of water. Thecontainer is rectangular in shape and is 5 cm wide, 7 cm long and 4 cm high. Find theconcentration of acid in the water, correct to two decimal places.

20 cm3

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ExplanationSolution

� 22.22%The concentration of acid is 22.22%.

Concentration �2090

� 100

� 70 cm3

V(water) �12

� 5 � 7 � 4 First find the volume of water

concentration (%) �V(acid)

total volume� 100

1IExercise

1 Find the total mass of these objects with the given densities, correct to one decimalplace where necessary.

a b c

2 Find the density of a compound with the given mass and volume measurements,rounding to two decimal places where necessary.

a mass 30 kg, volume b mass 10 g, volume c mass 550 kg, volume

3 Find the concentration of acid as a percentage if of pure acid is mixed into thegiven containers which are half full (half the volume) of water. Give your answerscorrect to two decimal places.

a b c

4 Find the volume of pure acid required to give the following concentrations and totalvolumes.

a concentration 25%, total volume b concentration 10%, total volume c concentration 32%, total volume 1.8 L

90 m3

100 cm3

6 cm

8 cm

2 cm

12 cm

5 cm

10 cm

10 cm3

1.8 m3

2 cm3

0.4 m3

6 cm

10 cm

5 cmDensity = 0.05 kg per cm3

2 m

Density =100 kg per m3

1 m

3 m

2 m

1 m

Density = 50 kg per m3

15Example

16Example

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5 Find the amount of space (volume)occupied by 100 kg of rock with adensity of correct to twodecimal places.

6 Five solid plastic spheres of radius 1.3 mare arranged to form an art piece. If thedensity of the plastic is find the total mass of the art piececorrect to two decimal places

7 A non-organic farmer mixes 5 litres of chemical weedicide into a three-quarter fulltank of water. The tank is cylindrical, with diameter 1 metre and height 1.5 metres.Find:

a the number of litres of water in the tank, correct to three decimal placesb the concentration of the weedicide after it has been added to the water, correct to

two decimal placesc what possible diameter and height measurements for the tanks would make the

concentration of acid 1%

25 kg per m3,

750 kg per m3,

37

Enrichment

8 The density of solids depends somewhat on how molecules are packed together.Molecules represented as spheres are tightly packed if they are arranged in a triangular form. The following relates to this packing arrangement.

a Find the length AC for three circles,each of radius 1 cm, as shown.Use exact values.

b Find the total height of four spheres,each of radius 1 cm, if they are packed toform a triangular-based pyramid.Use exact values.

First note that for an equilateral triangle (shown). Pythagoras’ theorem can be used to prove this, but this is difficult.

AB � 2BC

C

A1 cm

A

B

C

Th

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Esse

ntia

l M

athe

mat

ics

VEL

S P

roje

cts

W O R K I N G

1 Developing formulasProve the given formulas for the following shapes by showing mathematical steps. Use thedetails provided in the diagrams to help. The first one is completed for you.

For parts a to e you can assume only that the area of a rectangle is the product of thelength and width.For f and g you can assume only that the area of a circle is given by For h you can assume only that the area of a parallelogram is given by For part i you can use the area formulas of a rectangle and a triangle.a Parallelogram Proof

b Triangle (1) c Triangle (2)

d Rhombus e Kite

f Circle g Semicircle

h Trapezium (method 1) i Trapezium (method 2)

A � 12 (a � b)hA � 1

2 (a � b)h

A � 18 �d2A � 1

4 �d2

A � 12 xyA � 1

2 xy

A � 12 bhA � 1

2 bh

� bh� l � w

A � Area (rectangle)A � bh

A � bh.A � �r2.

MeasurementMathematically

b

h

h

yx

bb

hx

yxy

x

d

r

d

r

a

b

h

a

h

a

h

b

+

PL

Th

DLT

Comm

E

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Esse

ntia

l M

athe

mat

ics

VEL

S P

roje

cts

2 Packing spheresHeights of packing arrangementsUse the diagrams below to find the total height of these arranged spheres if each spherehas a radius of 1 cm. Joining the centres of the spheres gives the following diagrams.

A maximum number of spheresFind the maximum number of spheres able to be packed into a box 20 cm long, 10 cmwide and 10 cm high if the following packing styles are used. Note that spheres should notbe allowed to appear above the level of the top of the box.

a squareb square pyramidc triangular pyramid

Varying packing arrangementsFind the maximum number of spheres of radius 1 cm that can fit inside a box 20 cm long,10 cm wide and 10 cm high if a combination of packing styles are allowed to be used.

3 Designing milk cartonsMilk cartons are to be designed to hold 1 litre of milk. They are to bemade in the shape of a square-based prism with a triangular top, asshown.

Between and of spare air space must be designed intoeach carton and the carton must be at least 22 cm high.

1-litre cartonDesign the 1-litre milk carton and show your design by drawing a diagram markingall side measurements.

Restricted surface areaRepeat part 1 with the extra condition that the total surface area of the carton is to bebetween and 700 cm2.690 cm2

50 cm310 cm3

Triangular pyramidSquare pyramidSquare

h

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Chapter summary


Rev

iew Conversion of units

AreaSquareRectangleTriangleRhombusParallelogramTrapeziumCircle

Volume-

V(sphere) � 43�r3

V(cone) � 13�r2h

base

h

V(pyramid) � 13 area(base) � height

r

h

V(cylinder) � �r2h

heightcross-section

section) � heightV(prism) � area(cross

A � �r2

A � 12(a � b)h

A � bhA � 1

2xyA � 1

2bhA � lwA � l2

Volume

(10)3

(100)3

(1000)3

Area

(10)2

(100)2

(1000)2

Length

10

100

1000

Base unit

mm

cm

km

m××÷

LengthPerimeter is the distance around the outside of a shapeCircumference

Pythagoras’ theorem

Exact values (surds) can be used whenPythagoras’ theorem is applied in succession asin problems involving three dimensions.

Total surface area (TSA)TSA is the sum of all the areas of the outsidesurfaces of an object.

Cylinder

Sphere

Cone

Scientific applications

or

concentration (%) �volume of substance

total volume�

1001

mass � density � volumedensity �mass

volume

rs

TSA � �r2 � �rs

r

TSA � 4�r2

r

hh

r

r

2πr

TSA � 2�r2 � 2�rh

a2 � b2 � c2

C � 2�r � �d

dr

c

b

a

h

r

r

0521681774c01_p002_043.qxd 8/11/06 11:18 AM Page 40



Review

Multiple choice questions

1 The sum of and written using is

A B C D E2 The perimeter of this composite shape is closest to

A 22 cm B 33 cm C 66 cm

D 52 cm E 28 cm

3 A right-angled triangle has hypotenuse 3 metres and one other side 2.1 metres. Thelength of the remaining side is closest to

A 6.9 m B 4.6 m C 13.4 m D 2.1 m E 3.7 m4 A circular grain storage container of diameter 3 metres is placed on a block of land in

the shape of a parallelogram with base length 40 metres and height 13 metres. The areaof land not covered by the container is closest to

A B C D E5 A cylindrical can has a paper label glued to its curved surface. If the can has a radius of

3.5 cm and height 12.2 cm, the area of the label correct to one decimal place is

A B C D E6 The difference between the total surface area of a sphere with radius 2 cm and a cone

with radius 2 cm and slant height 6 cm is

A B C D E7 The volume of this composite solid is closest to

A B C

D E

8 The volume of air in a sphere is The radius of the sphere, correct to twodecimal places is

A 1.67 cm B 10.00 cm C 2.82 cm D 23.87 cm E 2.88 cm9 A meteorite has density 1300 kg/m3 and volume The mass of the meteroite is

A 0.00015 kg B 8667 kg C 195 kg D 1300 kg E 0.15 kg

1 Convert the given measurements to the units in the brackets.

a 0.23 m (cm) b c

d e 6.25 km (miles) f 0.0003 km2 (cm2)8.372 litres (cm3)

2.6 m3 (cm3)270 mm2 (cm2)

0.15 m3.

100 cm3.

94 m237 m2

30 m216 m29 m2

12.7 cm26 cm2� cm20 cm225 cm2

137.4 cm2549.8 cm2268.3 m2351.9 cm285.4 cm2

513 m2492 m2232 m2253 m2511 m2

26 m264 m2163 m2496 m23 700 260 001 m2

m2370 000 cm226 m20.0001 km2,


2 cm

7 cm

1 m

3 m

3 m

Short-answer questions

0521681774c01_p002_043.qxd 8/11/06 11:18 AM Page 41



Rev

iew

2 Find the perimeter of these shapes, correct to two decimal places, using Pythagoras’theorem where necessary.a b c

3 For a cuboid with dimensions 2, 4 and 7 units as shown, use Pythagoras’ theorem tofind:

a AF leaving your answer in surd form

b AG to two decimal places

4 Find the area of these shapes, correct to two decimalplaces.a b c

5 Find (i) the perimeter and (ii) the area of these composite shapes correct to two decimalplaces.a b c

6 Find (i) the total surface area and (ii) the volume of these solids correct to two decimalplaces where necessary.a b c

12 cm13 cm

10 cm

10 cm

4.6 mm

1.1 mm

18 cm

5

64

1.5 cm5 m

12 m

4.1 mm

1.5 mm1.6 mm

23.5 m2.3 cm

4.1 cm

1.37 mm

2.5 mm

3.17 mm

7 cm

10 cm

3 cm

3.75 m

2.6 m


H

D

AB

CE F

G2

7

4

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Review


7 a Find the mass of a plastic cylinder with radius 75 mm and height 22 cm if its densityis correct to three decimal places.

b A circular cone of radius 2.5 cm and height 12 cm is half filled (half the volume)with water. If of pure acid is mixed with the water in the cone, find the finalacid concentration expressed as a percentage, correct to one decimal place.

1 A water ski ramp consists of a rectangular floatation container and a triangular angledsection as shown.

a What volume of air is contained within the entire ramp structure?

b Find the length of the angled ramp (c metres), correct to 1 decimal place.The entire structure is to be painted with a waterproof paint costing $20 per litre.One litre of paint covers 25 square metres.

c Find the total surface area of the ramp, correct to one decimal place.

d Find the number of litres and the cost of paint required for the job. Assume you

cannot buy anything less than one litre of paint at a time.2 A circular school oval of radius 50 metres is marked with chalk to

form a square pitch as shown.

a State the diagonal length of the square.

b Use Pythagoras’ theorem to find the side of the square, correct

to three decimal places.

c Find the area of the square pitch.

d Find the percentage area of the oval not part of the square pitch. Round to the nearest

whole percent.

Two athletes challenge each other to a one-lap race around the oval. Athlete A runsaround the outside of the oval at an average rate of 10 m/s. Athlete B runs around theoutside of the square at an average rate of 9 m/s. Athlete B’s average running speed isless because of the need to slow down at each corner.

e Find who comes first, and the time difference, correct to the nearest hundreth of a

second.

5 m2 m 8 m

1 m

c m

10 cm3

8.5 kg per m3,

Extended-response questions

MC

TEST

D&D

TEST

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