radians, arc length and sector area

40: Radians, Arc Length 40: Radians, Arc Length and Sector Areaand Sector Area

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Radians, Arc Length and Sector Area

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Module C2


Radians

Radians are units for measuring angles.They can be used instead of degrees.

r

O

1 radian is the size of the angle formed at the centre of a circle by 2 radii which join the ends of an arc equal in length to the radius.

r

r

x = 1 radian

x

= 1 rad. or 1c


r

O

2r

r

2c

If the arc is 2r, the angle is 2 radians.

Radians


O


r3r

r

3c


Radians


O


c143


r

r

If the arc is r, the angle is radians.

143 143

r143

Radians


O


r

r



143 143


rc

Radians



O

r

r

rc

But, r is half the circumference of the circle so the angle is

180

180 radians Hence,

Radians


We sometimes say the angle at the centre is subtended by the arc.

180 radians

Hence,

180

357

radian 1

r

O

r

rx

x = 1 radian357

Radians


Radians

SUMMARY

• One radian is the size of the angle subtended by the arc of a circle equal to the radius

180 radians •

• 1 radian 357


Exercises

1. Write down the equivalent number of degrees for the following number of radians:

Ans:

(a) (b) (c) (d)2

3

26

(a) (b) (c) (d)60 45 120 30

2. Write down, as a fraction of , the number of radians equal to the following:

(a) (b) (c) (d)6090 360 30

(a) (b) (c) (d)3

6

32

4

Ans:

It is very useful to memorize these conversions


Arc Length and Sector Area

Let the arc length be l .

O

r

r

l

rl 22

Consider a sector of a circle with angle .

θ

Then, whatever fraction is of the total angle at O, . . .

θ

θrl

2

θ. . . l is the same fraction of the circumference. So,

( In the diagram this is about one-third.)

2

l circumference

2

lcircumference


O

r

r

θ

θrA 2

2

1

Also, the sector area A is the same fraction of the area of the circle.

A

2

Acircle area

2

2rθA



Examples

1. Find the arc length, l, and area, A, of the sector of a circle of radius 7 cm. and sector angle 2 radians.

Solution: where is in radians

θrl θ

cm.14)2)(7( ll

θrA 221 .cm249)2()7(

2

1 2 AA


2. Find the arc length, l, and area, A, of the sector of a circle of radius 5 cm. and sector angle . Give exact answers in terms of .

150

Solution: where is in radians

θrl θ180 rads.

630

rads

. 6

5150

rads.

So, cm.6

25

6

55

llrθl

θrA 221 .cm2

12

125

6

5)5(

2

1 2

AA

Examples


Radians• An arc of a circle equal in length to the

radius subtends an angle equal to 1 radian. 180 radians •

• 1 radian 357

θrl

θrA 221

For a sector of angle radians of a circle of radius r,

θ

• the arc length, l, is given by

• the sector area, A, is given by

SUMMARY


1. Find the arc length, l, and area, A, of the sector shown. O

4 cm

A

c2

l

2. Find the arc length, l, and area, A, of the

sector of a circle of radius 8 cm. and sector

angle . Give exact answers in terms of

.

120

Exercises


1. Solution:

θrl cm.8)2)(4( l

θrA 221 .cm216)2()4( 2

21 A

O4 cm

A

c2

l

Exercises


2. Solution:

180 rads.

360

rads

.

3

2120

rads.

So, cm.3

16

3

28

llrθl

θrA 221 .cm2

3

64

3

2)8(

2

1 2

AA

O8 cm

A

120

l

where is in radians

θrl

Exercises


The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.


Radians

SUMMARY

• One radian is the size of the angle subtended by the arc of a circle equal to the radius

180 radians •

• 1 radian 357

r

O

r

rx

Radians, Arc Length and Sector AreaArc Length and Sector

Area

Let the arc length be l .

O

r

r

l

rl 22

Consider a sector of a circle with angle .

θ

Then, whatever fraction is of the total angle at O, . . .

θ

θrl

2

θ. . . l is the same fraction of the circumference. So,

( In the diagram this is about one-third.)

2

l circumference

2

lcircumference


2

2rθA


O

r

r

θ

θrA 2

2

1

Also, the sector area A is the same fraction of the area of the circle.

A

2

Acircle area


SUMMARY

θrl

θrA 221

For a sector of angle radians of a circle of radius r,

θ

• the arc length, l, is given by

• the sector area, A, is given by

radians, arc length and sector area

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