- One radian is the size of the angle subtended by the arc of a circle equal to the radius. 1 radian is approximately 57 degrees. - For a sector of angle θ radians of a circle of radius r, the arc length l is given by l = θr/2π and the sector area A is given by A = (θr^2)/2.

1. 40: Radians, Arc Length and40: Radians, Arc Length and
Sector AreaSector Area
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core ModulesVol. 1: AS Core Modules

2. Radians, Arc Length and Sector Area
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Module C2

3. Radians, Arc Length and Sector Area
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 1
c

4. Radians, Arc Length and Sector Area
r
O
2r
r
2c
If the arc is 2r, the angle is 2 radians.
Radians

5. Radians, Arc Length and Sector Area
O
If the arc is 3r, the angle is 3 radians.
r
3r
r
3c
If the arc is 2r, the angle is 2 radians.
Radians

6. Radians, Arc Length and Sector Area
O
If the arc is 3r, the angle is 3 radians.
c
143⋅
If the arc is 2r, the angle is 2 radians.
r
r
If the arc is r, the angle is radians.143⋅ 143⋅
r143⋅
Radians

7. Radians, Arc Length and Sector Area
O
If the arc is 3r, the angle is 3 radians.
r
r
If the arc is 2r, the angle is 2 radians.
If the arc is r, the angle is radians.143⋅ 143⋅
If the arc is r, the angle is radians.π π
πrc
π
Radians

8. Radians, Arc Length and Sector Area
If the arc is r, the angle is radians.π π
O
r
r
π rc
π
But, r is half the circumference of the circle
so the angle is
π

180

180radians =πHence,
Radians

9. Radians, Arc Length and Sector Area
We sometimes say the angle at the centre
is subtended by the arc.

180radians =π
Hence,
π

180

357⋅≈
radian =1
r
O
r
rx
x = 1 radian

357⋅≈
Radians

10. Radians, Arc Length and Sector Area
 Radians
SUMMARY
• One radian is the size of the angle subtended
by the arc of a circle equal to the radius

180radians =π•
• 1 radian ≈ 
357⋅

11. Radians, Arc Length and Sector Area
Exercises
1. Write down the equivalent number of degrees
for the following number of radians:
Ans:
(a) (b) (c)
(d)2
π
3
π π2
6
π
(a) (b) (c)
(d)

60 
45 
120 
30
2. Write down, as a fraction of , the number of
radians equal to the following:
π
(a) (b) (c) (d)
60
90 
360 
30
(a) (b) (c) (d)3
π
6
π
3
2π
4
π
Ans:
It is very useful to memorize these conversions

12. Radians, Arc Length and Sector Area
Arc Length and Sector Area
Let the arc length be
l .
O
r
r
l
rl π
π
θ 2
2
×=⇒
Consider a sector of a circle with angle .θ
Then, whatever fraction is of
the total angle at O, . . .
θ
θrl =⇒
π2
θ
. . . l is the same fraction of the
circumference. So,
( In the diagram this is about one-third.)
×=
π
θ
2
l circumference⇒
π
θ
2
=
l
circumference

13. Radians, Arc Length and Sector Area
O
r
r
θ
θrA 2
2
1
=⇒
Also, the sector area A is the same
fraction of the area of the circle.
A
π
θ
2
=
A
circle area
π
π×
=
2
2
rθ
A⇒
Arc Length and Sector Area

14. Radians, Arc Length and Sector Area
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 radiansθrl = θ
cm.14)2)(7( =⇒=⇒ ll
θrA 2
2
1= .cm2
49)2()7(
2
1 2
=⇒=⇒ AA

15. Radians, Arc Length and Sector Area
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
π
Solution: where is in radiansθrl = θ

180rads. =π
6
30
π
=⇒  rads.
6
5
150
π
=⇒  rads.
So, cm.
6
25
6
5
5
π
=⇒
π
×=⇒= llrθl
θrA 2
2
1= .cm2
12
125
6
5
)5(
2
1 2 π
=⇒




 π
=⇒ AA
Examples

16. Radians, Arc Length and Sector Area
 Radians
• An arc of a circle equal in length to the
radius subtends an angle equal to 1 radian.

180radians =π•
• 1 radian ≈ 
357⋅
θrl =
θrA 2
2
1=
 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

17. Radians, Arc Length and Sector Area
1. Find the arc length, l,
and area, A, of the
sector shown.
O
4 cm
A
c
2
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
π
Exercises

18. Radians, Arc Length and Sector Area
1. Solution:
θrl = cm.8)2)(4( ==⇒ l
θrA 2
2
1= .cm2
16)2()4( 2
2
1 ==⇒ A
O
4 cm
A
c
2
l
Exercises

19. Radians, Arc Length and Sector Area
2. Solution:

180rads. =π
3
60
π
=⇒  rads.
3
2
120
π
=⇒  rads.
So, cm.
3
16
3
2
8
π
=⇒
π
×=⇒= llrθl
θrA 2
2
1= .cm2
3
64
3
2
)8(
2
1 2 π
=⇒




 π
=⇒ AA
O
8 cm
A

120
l
where is in radiansθθrl =
Exercises

20. Radians, Arc Length and Sector Area

21. Radians, Arc Length and Sector Area
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.

22. Radians, Arc Length and Sector Area
 Radians
SUMMARY
• One radian is the size of the angle subtended
by the arc of a circle equal to the radius

180radians =π•
• 1 radian ≈ 
357⋅
r
O
r
rx

23. Radians, Arc Length and Sector Area
Arc Length and Sector Area
Let the arc length be
l .
O
r
r
l
rl π
π
θ 2
2
×=⇒
Consider a sector of a circle with angle .θ
Then, whatever fraction is of
the total angle at O, . . .
θ
θrl =⇒
π2
θ
. . . l is the same fraction of the
circumference. So,
( In the diagram this is about one-third.)
×=
π
θ
2
l circumference⇒
π
θ
2
=
l
circumference

24. Radians, Arc Length and Sector Area
π
π×
=
2
2
rθ
A
Arc Length and Sector Area
O
r
r
θ
θrA 2
2
1
=⇒
Also, the sector area A is the same
fraction of the area of the circle.
A
π
θ
2
=
A
circle area
⇒

25. Radians, Arc Length and Sector Area
SUMMARY
θrl =
θrA 2
2
1=
 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