8. Circ Measure

7/29/2019 8. Circ Measure

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CIRCULAR MEASURE

ADDITIONAL

MATHEMATICS

MODULE 15

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CONTE NTS PAGE

8.0 Concept map 2

8.1 Radian 3

8.2 Arc length of a circle 3

8.3 Area of sector of a circle 5

8.4 SPM Questions 7

8.5 Assessment 10

8.6 Answers 14

CHAPTER 8: CIRCULAR MEASURE



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CHAPTER 8 : CIRCULAR MEASURE

8.0 CONCEPT MAP

Circular Measure

s

r

s

r

Area of sector, A

A =

Arc length, s

s =

R

Q

O

Area of segment RQ

= area of sector ROQ area ofORQ

=

=

0180rad

Radian Degree Degree Radian



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8.1 Convert Measurements in radians to degrees and vice versa.

8.1 a Convert the following angles in radians to degrees and minutes.

a. 1.5 rad b. 0.63 rad

c. rad2

d. rad

2

3

8.1b Convert the following angles to radians.

a. 50 b. 124.3

c. 72 35 d. 285 21

8.2 Arc Length of a circle8.2a Find the length of arc.

1.

0.5 rad

8 cm

Q

P

O

2.

152

6.4 cmO

BA



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8.2b Complete the table below by finding the values of , r or s.

r s

1. 1.5 rad 9 cm

2. 14 cm 30 cm

3. 2.333 rad 35 cm

8.2c Find the perimeter of each of the shaded segment below.

1

5 c m O S

RQ

P

1.2 rad

2.

85

O

S

Q

P

R

Given

OP = 5cm

the length of

RQ = 3 cm



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8.3 Area of sector of a circles8.3a Find the area of each of the shaded sectors below.

1.

705 cm

O

2.

10 cm

120

O

3.

rad

O

3

15 cm

4.

O

250

20 cm



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8.3b Complete the table below, given the areas and the radii of the sectors andangles subtended.

Area of sector Radius Angle subtended

1. 90 cm 9.15 cm

2. 38.12 cm 50

3. 72 cm =1.64 rad

4. 18cm2 6.5 cm

5. 200 cm 1.778 rad

6. 145 cm 8 cm



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8.3c Find the area of each shaded segment below.

a.

O

Q

P

50

7 cm

b.

4.8 cm

6.4 cm

G

F

O

8.4 SPM QUESTIONS

1.

0.35 radO

B

A

Diagram above shows a circle with centre

O. Given that the length of the major arcAB is 45.51 cm, find the length, in ,cm, of

the radius. (Use = 3.142).[3 marks](Paper 1,2004)

2.

S

R

O

Diagram above shows a sector ROS with

centre O. The length of the arc RS is 7.24

cm and the perimeter of the sector ROS is

25 cm. Find the value of in rad.

[3 marks](Paper 1, 2003)



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3.

O

B

A

Diagram above shows a circle with centre O. The length of the minor arc AB is 16 cm

and the angle of the major sector AOB is 2900. Using = 3.142, find

a. the value of , in radians,

(Give your answer correct to four significant figures.)

b. the length, in cm,of the radius of the circle. [ 3 marks]

(Paper 1,2005)



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4.

QR

P

O

Calculate

a. the value of ,in rad [ 3 marks]

b. the area of shaded region, in cm2

[4 marks]

(Paper 2, 2003)

5.

8 cm

rad

6Q

P

B

A

O

Diagram shows a sector POQ of a circle,

centre O. The point A lies on OP, the point

B lies on OQ and AB perpendicular to OQ.The length of OA = 8 cm and POQ =

radian6

It is given that OA : OP = 4 : 7.

(Use = 3.142). Calculate

a. the length, in cm,of AP [1 mark]

b. the perimeter, in cm, of the shadedregion, [5 marks]

c. the area, in cm2, of the shaded region.

[4 marks]

(Paper 2 ,2005)

The diagram showsthe sector POQ,

centre O with radius

10 cm. The point Ron OP is such thatOR:OP = 3:5.



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6.

rad

T

O

K

L

J

RQ

P

(Paper 2,2004)

Diagram shows a circle PQRT, centre O and radius 5 cm.

JQK is a tangent to the circle at Q. The straight lines, JO andKO, intersect the circle at P and R respectively. OPQR is a

rhombus. JLK is an arc of a circle, centre O. Calculate

a. the angle

, in terms of

[ 2marks

]b. the length, in cm, of the arc JKL [ 4 marks]

c. the area, in cm2, of the shaded region. [4 marks]



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TESTYOURSELF

1.

D

C

BA

2.

9 cm

O

D

B

A

3.

1.4 rad 7 c mO

D

C

B

A

4.

30

O

C

BA

The diagram shows,A and B are the

centres of two

circles, each withradius 6 cm.

a. Find CBD in radians

b. Show that the area of the shaded

region is 137.7 cm2

In the diagram, ABand DB are tangents

to the circle with

centre O and radius9 cm. If ABD =

750

calculate

a. AOD in radius

b. the area of the shaded region

The diagram shows a

sector circle. Given

that OA : OB = 3:2,find

a.The length of OA

and AB

b. the perimeter of theshaded region

The diagram shows acircle with centre O.

Given the radius of thecircle is 15 cm. Find,a. the area of sector

OAB.

b. the area of segmentAB



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8.5 ASSESSMENT (30 MINUTES)

1.

Q

P

O

2.

3045

O

DE

B

A

Diagram shows a circle

with centre O anddiameter 16 cm. Given

the length of arc PQ is

16 cm. Find

a. , in degree and minutes

b. the length of PQ

c. the area of the shaded region

Diagram shows two

arc, AE and BD,subtended the

angles of 300

and45

0respectively at

centre O.

ABO is a straight line and B is the

midpoint of OA. Given that OD = 6 cm,

finda. the perimeter of the whole diagram

b. the area of the whole diagram

c. the area of the shaded region.



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3.

8.4 cm1.25 rad

4.

0.75 cm 2.4 cm

6.4 cm

O

CB

A

Mr.Lee wishes

to fence a place

shaped like asector of a circle

as shown in

diagram. Find

a. the area of the sector

b. the total length of the fence needed

by Mr. Lee

In diagram, O is

the centre of thecircle. Given the

length of arc AB

and BC are 2.4 cmand 6.4 cm

respectively and

AOB = 0.75

rad. Calculate

a. the length of radius OC

b. BOC



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8.1 a

a. 85057 b. 3606c. 90

0d. 270

0

8.1b

a. 0.87 rad b. 2.17 radc. 1.27 rad d. 4.98 rad

8.2a

1. 4 cm 2. 16.98 cm

8.2b

1. s = 13.5 cm 2. 2.14 rad

3. r = 15 cm

8.2c

1. 29.708 cm 2. 19. 56 cm8.3a

1. 15.27 cm2

2. 104.72 cm2

3. 117.8 cm2

4. 383.97 cm2

8.3b

1. rad15.2 2. A = 634.05 cm2

3. r = 9.34 cm 4. rad68.2

5. r = 15 cm 6. rad53.48.3c

a. 2.548 cm2

b. 1.536 cm2

TEST YOURSELF

1 a. 2.094 rad

2. a. 1.833 rad

b. 31.34 cm2

3. a. OA = 3 cm

AB = 2 cm

4. a. 117.8 cm2

b. 20.38 cm2

SPM QUESTION

1. j = 7.675 cm

2. = 0.8153 rad

3. a. = 1.222 rad

b. j = 13.09 cm

4. a. 0.9237 radb. 22.365 cm

2

5. a. AP = 6 cm

b. Perimeter = 24.403 cmc. Area of the shaded region

= 37.46 cm2

6. a. 32 rad

b. 20.94 cm

c. 61.40 cm

ASSESSMENT

1. a. 1140

35b. Length of PQ = 13.462 cm

c. Area of shaded region = 137.067 cm2

2. a. 34.997 cmb. 51.843 cm

2

c. 19.704 cm2

3. a. 28.224 m2

b. 21.84 m4. a. 3.2 cm

b. 2 radian

ANSWER

8. Circ Measure

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