UNIT 5

UNIT 5 : CIRCULAR MEASURE

5.1 Degrees and Radians

LEARNING OUTCOMES Angle in circle can be measured in either in degree (o) or radian (rad).
1 radian is obtained when the value of arc length is equal to the length of
After completing the unit, students radius in a circle.
should be able to:
Relationship between degree and radian
1. Convert angles from
degree to radian and 3600 = 2 rad Central angle
vice versa. 1800 = rad 

2. Determine circular 10 = 6π0’ radius, r
Measurement as
arc length, area of 10 =
sector and area of
segment of circle. 180

1rad = 180
π

Figure 5.2

Example 5.1:

Convert each angle given to radian:

a) 600
b) 58.30

Solution: b) 58.30  π
1800
a) 600  π
1800 = 1.02 rad

= 1.047 rad

PRESS MODE RAD 2 Unit 5: Circular Measures
60 SHIFT DRG 1
= 1.047

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UNIT 5 : CIRCULAR MEASURE

Example 5.2 :

Convert each angle given to degree:

a) 0.84 rad

b) 3 rad
2

Solution:

a) 0.84  1800



= 48.130

PRESS MODE RAD 1

0.84 SHIFT DRG 2 = 48.13

b) 3  1800

2

= 85.940

Practice 5.1:
1. Change the following angles in terms of radians:

a. 48o b. 8.50

c. 520 21’ d. 620 9’

e. 123.40 f. 4500 Unit 5: Circular Measures

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UNIT 5 : CIRCULAR MEASURE

2. Change the following angles into degrees

a. 0.8 radian b. 4 π radian
c. 1.24 radian 3
π
e. 1 1 π radian
3 d. radian
2

f. 2 1 radian
4

g. 2.56 radian h. 1 radian

5.2 Arc Length

Consider a circle of radius r with centre O as shown in Figure 5.2. Given θ is the angle
subtended at the centre by the minor arc length.

Minor arc length, s

Major arc length

radius, r

Figure 5.2 Unit 5: Circular Measures

Arc length, s = rθ
where;

r = radius, θ = angle in radian

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UNIT 5 : CIRCULAR MEASURE

Example 5.3 :

5 cm s

1.2 rad
O

Figure 5.3

Figure 5.3 shows a circle with center O, and radius 5 cm. Given angle θ = 1.2 rad. Find the
value of arc length, s.

Solution :

s = rθ

= 51.2
= 6cm
Example 5.4 :

10 cm A
O

720

s
B

Figure 5.4 Unit 5: Circular Measures

Figure 5.4 shows a circle with center O. Given radius = 10 cm and angle AOB = 720. Find
the value of arc length, s.

Solution :

Convert angle in degree to radian,

θ = 720  π
1800

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UNIT 5 : CIRCULAR MEASURE

= 1.26 rad

Then,
s=rθ

= 10 × 1.26 rad

= 12.6 cm

Example 5.5 :

A
r
O 2.6 rad 18.2 cm

B

Figure 5.5

Figure 5.5 shows a circle with center O. Given angle AOB = 2.6 rad and arc length,

s of AB = 18.2 cm. Find the value, r.

Solution:
s=rθ

r = s
θ

= 18.2
2.6

= 7 cm

Unit 5: Circular Measures

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UNIT 5 : CIRCULAR MEASURE

Practice 5.1:

1. Figure 5.3 shows a circle with center O. Evaluate
the value of the arc length of the minor sector
O A AOB if angle AOB = 2 rad and radius OA = OB =
7 cm 2 rad 7cm.

B Figure 5.4 shows a circle with center O,
radius = 5cm and an angle of NOM = 1000. Find
Figure 5.3 the arc length of the minor sector NOM.

2.
N
M
1000

5 cm
O

Figure 5.4

3. Figure 5.5 shows a circle with a center O. Find
the radius of OB if given the arc length of minor

A sector AOB = AOB = 28 cm and the angle of
O 0.7rad AOB = 0.7 radian.

28 cm

B Unit 5: Circular Measures
B

Figure 5.5

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UNIT 5 : CIRCULAR MEASURE Unit 5: Circular Measures

4.
a) Find arc of length, s if r = 7 cm and  = 30o .
b) Find arc of length, s if r = 20 cm and  =  rad.

3

c) Find radius, r if s = 5cm and  = 0.6 rad.

d) Find radius, r if s = 8cm and  = 2.5 rad.

e) Find angle of  if r =9 cm and s = 20 cm.

f) Find angle of  if r =5 cm and s =  .

82

UNIT 5 : CIRCULAR MEASURE Figure 5.6 shows a sector STU formed from a
piece of a wire with a centre T. If the length of
5. S the wire is 50 m, and the radius of TS = TU = 7
m, find the angle, θ of the sector STU in:
7m a) radian
Tθ
b) degree.
U
Figure 5.6

6. A The figure 5.7 shows a sector OAB of a circle,
5cm Q centre O, formed from a piece of wire of length
19 cm. Given that OA = OB = 5cm, find the angle

θ , in radians.

θ

O
5cm

B

Figure 5.7

Unit 5: Circular Measures

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UNIT 5 : CIRCULAR MEASURE

5.3 Area of Sector

Minor sector 1
Area of sector, A = 2 r2 θ
Major sector 
radius, r Where ;
r = radius, θ = angle in

radian

Figure 5.8 r = radius, θ = angle in
Figure 5.7 radian

Example 5.7 :

P
10 cm
O 720

R

Figure 5.8 Unit 5: Circular Measures

Figure 5.8 shows a circle with center O, and radius = 10 cm. Given an angle
POR = 720. Find the area of the shaded region.

Solution:

θ = 720  π
1800

= 1.26 rad

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UNIT 5 : CIRCULAR MEASURE

A = 1 r2θ
2

= 1 102 1.26
2

= 63 cm 2

Example 5.8 :

a. b. S
4.8rad T
7 cm X

Y 00.5.5raradd

6 cm U
Z

Figure 5.9

Calculate the area of the shaded region above.

Solution :

a. Area of the shaded region b. Area of the shaded region

= 1 r2θ = 1 r2θ
2 2

= 1 (7)2 (0.5) = 1 (6)2 (4.8)

2 2
= 12.25 cm2 = 86.4 cm2
Note:Minor sector Note:Major sector

Unit 5: Circular Measures

85

UNIT 5 : CIRCULAR MEASURE Figure 5.9 shows a circle with center O, and radius =
8 cm. Given the angle of a minor sector is 0.4
Practice 5.3 : radian, find the value of the shaded region.

1.
8 cm

O 0.4 rad

Figure 5.9 Find the area of sector that 20o subtended at the
center of the circle of radius 5 cm. Give your answer
2. correct to two decimal places. (Take  = 3142)

5 cm
20o

Figure 5.10

Figure 5.11 shows a circle POQ with a center O,

3. radius OP = OQ = 4.5 cm and an angle of
Q
POQ = 1100. Evaluate:
P 110 0 M
4.5 cm O a) the value of an angle in radian

Figure 5.11 b) the area of minor sector POQ

c) the area of major sector POQ

Unit 5: Circular Measures

86

UNIT 5 : CIRCULAR MEASURE The Figure 5.12 shows a sector OAB of a circle,
centre O and radius 7 cm. given that the area of
4. A
sector OAB is 12.83 cm2. find  AOB in degree.
7 cm
O

B The Figure 5.13 shows two arcs, AB and CD, of two
concentric circles, center O, with radius OA and OC
Figure 5.12 respectively. Given that the length of arc AB = 6 cm
, the length of arc CD = 2 cm and the length of OA
B = 12 cm. find
5. C Q
a) the angle  in radians,

b) area of the shaded region, ABCD.
O

D
A

Figure 5.13

6. . Figure 5.14 show a part of the circle with centre O.
R a) Calculate the values of r,
b) Find the angle QOP in degree.
Q

r 12cm

1.2rad P
O
Unit 5: Circular Measures
Figure 5.14

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UNIT 5 : CIRCULAR MEASURE

7. Figure 5.15 shows two circle with centre O. Given OB
= 4 cm, BD = 8 cm and CD = 8.376 cm
C D a) Find the angle OAB in degree and radian,
A B
O b) Calculate the perimeter of the shaded portion
ABCD,

c) Calculate the area of the shaded portion
ABCD.

Figure 5.15

8. Figure 5.16 shows two sector with centre O. Given
OB = 5 cm, PB = 4 cm and the area of the sector
P POQ is 15cm2.
B
O a) Find the angle AOB in radian,

AQ b) Calculate the area of the shaded region,
Figure 5.16
c) Calculate the perimeter of the shaded region.

Unit 5: Circular Measures

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UNIT 5 : CIRCULAR MEASURE

5.4 Area of Segment

A

Oθ Segment Area of segment = area of sector AOB – area
of triangle AOB

= 1 r 2θ − 1 r 2 sinθ
22

( )B = 1 r 2  − sin
2

Figure 5.15 1 – angle in radian
Example 5.10 : 2 – angle in degree

Figure 5.12 shows a roundabout with center O, and radius 10 cm. Given an angle AOB of
the roundabout = 720. Find the area of segment AOB.

Solution:

A

10 cm

 = O 72o
1800
x720

= 1.26 rad B
Figure 5.12

Area of segment of AOB 1 r2 ( θ1 – sin θ2 )
=

2

= 1 X 102 X (1.26 – sin 720) Unit 5: Circular Measures
2

= 15.45 cm2

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UNIT 5 : CIRCULAR MEASURE A

Practice 5.4 : 10 cm

1. The Figure 5.17 shows a circle with radius 10 cm and  AOC = 1.4 rad. O 1.4 rad
a) Convert  AOC in degree.
Figure 5.17
b)Find the area of the segment AOC C

2. Figure 2 shows a sector with radius 8 cm and angle 60.2o. B
Find the area of the segment. A

8 cm
60.2o

O

3. Figure 5.18 shows the sector AOB with radius 6 cm B
and centre O. Given the length of arc AB = 4.2 cm.
a. Find the angle AOB in degree and radian, O
A
b. Calculate the area of the shaded region.
Figure 5.18

Unit 5: Circular Measures

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UNIT 5 : CIRCULAR MEASURE

4 cm P 4) Figure 5.19 shows a sector with center O, and radius 4 cm.
O 24 cm Given arc length of PQ = 24 cm.
a) Find the angle POQ in radian and degree,

Q b) Find the area of the shaded region.
Figure 5.19

O 5) Given the value of radius is 11 cm and  θ = 1250, find
11 cm θ
Q a) angle in radian

P b) the area of segment POQ.

Unit 5: Circular Measures

91