Circular Measure

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Chapter 8- Circular Measure

Additional Mathematics Module Form 4 SMK Agama Arau, Perlis

CHAPTER 8- CIRCULAR MEASURE

8.1 RADIAN 1. In lower secondary, we have learned the unit for angle is degree. In this chapter, we will learn one more unit for angle that is radian. P r O 1 radian r Q 2. When the value of the angle 1 radian, then the length of the arc is equal to the length of the radius. 3. From this information, we can deduce that: r

1 rad r = 360 2πr
1 rad = r 2πr × 360

2π rad = 360°
4. π rad = 180°

1 rad =

180°

π

= 57.3° or 57°18 '
5. 2π rad = 360°

1° =

π
180

radian

8.1.1 Converting Measurements in degree to radian Example 1: Convert 120° to radians Solution:

1° =

π
180

radian

Page | 94

Chapter 8- Circular Measure

Additional Mathematics Module Form 4 SMK Agama Arau, Perlis

120° = 120 × =

π
180

radian

2π radian or 2.0947 radian 3
Tips

Example 2: Convert 112°36 ' to radians Solution:

1° = 60'

112°36 ' = 112° + 36 ' 36 = 112° + ( )° 60 = 112° + 0.6° = 112.6° π 1° = radian 180 π 112.6° = 112.6 × radian 180 = 1.965 rad 8.1.2 Converting Measurements in radian to degree Example 1: Convert

π
6

rad to degree

Solution:

1 rad =

180°

π π π 180° rad = × 6 6 π
= 30°

Example 2: Convert 1.36rad to degree Solution:

1 rad =

180°

π
Page | 95

Chapter 8- Circular Measure

Additional Mathematics Module Form 4 SMK Agama Arau, Perlis

1.36rad = 1.36 × = 244 °

180°

π

π

= 77.92°
EXERCISE 8.1 1. Convert each of the following values to degrees and the nearest minute. (π = 3.142 ) (a) 0.37 rad (b) 2.04 rad (c) 1.19 rad 2. Convert each of the following values to radians, giving your answer correct to 4 significant figures.



= 3.142)

(a) 248°9 ' (b) 304°22 ' (c) 46°14 ' 8.2 LENGTH OF ARC OF A CIRCLE P S

θ
O Q

S Circumference of a circle

=

θ
angle of a whole turn

If the unit of angle is degrees, The angle of a whole turn is 360° .

S θ = 2πr 360 °

θ S= × 2πr 2π
We know that 2π rad = 360° If the unit of the angle is in radians, The angle of a whole turn is 2π radian.

θ is in degrees.

Page | 96

Chapter 8- Circular Measure

Additional Mathematics Module Form 4 SMK Agama Arau, Perlis

Hence,

S Circumference of a circle

=

θ 2π

θ S = 2πr 2π 2πrθ S= 2π S = rθ
Example 1: P

θ is in radians.

7.68cm

θ
O
Q

The diagram above shows a circle with a sector POQ and radius 6 cm. Given the length of minor arc PQ is 7.68 cm. Find the value of θ , in radians. Solution: The formula of length of arc if angle in radians is

S = rθ
Given r = 6 cm and S= 7.68 cm,

7.68 = 6θ

θ = 1.28 rad
Example 2: Given a circle with centre O and radius 5 cm. Find the length of arc PQR if the angle ∠POR is 1.2 rad. Solution: The formula of length of arc if angle in radians is

S = rθ
Given r = 5 cm and θ = 1.2 rad,

S = (5)(1.2) S = 6 cm
Page | 97

Chapter 8- Circular Measure

Additional Mathematics Module Form 4 SMK Agama Arau, Perlis

EXERCISE 8.2 1. In the diagram 3, the perimeter of sector OPQ is 32 cm. P

θ
O Q Diagram 3 (a) Express r in terms of θ . (b) Find the value of r if θ = 1.2 rad. 2. The angle subtended at the centre by an arc ABC with a radius 4.2 cm is 1.4 radians. Find the length of arc AB. 3. The length of a minor arc of a circle is 2π cm. The angle subtended at the centre of by the major arc is 240° . Find the radius of the circle. 4. Diagram 4 shows two arcs AB and CD with a common centre O. It is given that BD= AC= 3cm . C

A

1 rad 3

O

B

D

Diagram 4 If the perimeter of the shaded region ABCD is 12 cm, find the length of radius OB.

8.3 AREA OF SECTOR OF A CIRCLE A

θ
O B

A Area of circle

=

θ
angle of a whole turn

Page | 98

Chapter 8- Circular Measure

Additional Mathematics Module Form 4 SMK Agama Arau, Perlis

If the unit of angle is degrees, The angle of a whole turn is 360° .

A θ = 2 360 °...
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