C2: Arcs, Sectors and
Segments
Learning Objective: to use radian
measure to calculate an arc
length, area of a sector and
segment of a circle
Starter:
1.
2.
3.
4.
5.
6.
7.
8.
360o =
180o =
90o =
270o =
45o =
60o =
120o =
30o =
Using radians to measure arc length
Suppose an arc AB of a circle of radius r subtends an angle of
θ radians at the centre.
If the angle at the centre is 1 radian
then the length of the arc is r.
A
If the angle at the centre is 2 radians
r
then the length of the arc is 2r.
θ
B
If the angle at the centre is 0.3 radians
r
O
then the length of the arc is 0.3r.
In general:
Length of arc AB = θr
where θ is measured in radians.

× 2 r
When θ is measured in degrees the length of AB is
360
Finding the area of a sector
We can also find the area of a sector using radians.
Again suppose an arc AB subtends an angle of θ radians at
the centre O of a circle.
The angle at the centre of a full circle
A
is 2π radians.
r
θ
O
r
B
So the area of the sector AOB is
of the area of the full circle.

2

× r2
 Area of sector AOB =
2
1
In general: Area of sector AOB = 2 r2θ
where θ is measured in radians.

× r2
When θ is measured in degrees the area of AOB is
360
Finding arc length and sector area
A chord AB subtends an angle of 23 radians at the
centre O of a circle of radius 9 cm. Find in terms of π:
a) the length of the arc AB.
b) the area of the sector AOB.
a) length of arc AB = θr
A
=
2
3
O
9 cm
B
2
×9
3
= 6π cm
b) area of sector AOB = 21 r2θ
1 2 2
= ×9 ×
2
3
= 27π cm2
Task 1:
1. An arc AB of a circle, centre O and radius r
cm, subtends an angle θ radians at O. Find the
length of the arc AB and the area of the sector
AOB when
r = 8cm, θ = 0.85
r = 3.5cm, θ = 0.15
r = 6cm, θ = 3/8 π
2. A sector of a circle of radius r cm contains an
angle of 1.2 radians. Given that the sector has
the same perimeter as a square of area 36
cm2 , find the value of r. Hence calculate the
area of the sector.
3. The area of a sector of a circle of radius 12cm
is 100 cm2. Find the perimeter of the sector.
Finding the area of a segment
The formula for the area of a sector can be combined with the
formula for the area of a triangle to find the area of a segment.
For example:
A chord AB divides a circle of radius 5 cm into two segments.
If AB subtends an angle of 45° at the centre of the circle, find
the area of the minor segment to 3 significant figures.
o
45 =
A
O
45°
5 cm
B

radians
4
Let’s call the area of sector AOB AS
and the area of triangle AOB AT.
AS =
1
2
r 2
= 21 × 52 × 4
= 9.8174... cm2
Finding the area of a segment
AT =
1
2
r 2 sin
= 21 × 52 × sin 4
= 8.8388... cm2
Now:
Area of the minor segment = AS – AT
= 9.8174… – 8.8388…
= 0.979 cm2 (to 3 sig. figs.)
In general, the area of a segment of a circle of radius r is:
A = 21 r 2 (  sin )
where θ is measured in radians.
Task 2:
1.
Calculate the area of the shaded segment
A
O
2.
30°
8 cm
B
Calculate the area of the segment when r = 12cm and
θ=π/4
Examination-style question
In the following diagram AC is an arc of a circle with centre O
and radius 10 cm and BD is an arc of a circle with centre O
and radius 6 cm.
AOD = θ radians.
A
a) Find an expression for the
area of the shaded region in
terms of θ.
B
6 cm
θ
O
10 cm
D
C
b) Given that the shaded region
is 25.6 cm2 find the value of θ.
c) Calculate the perimeter of the
shaded region.
Examination-style question
a) Area of sector AOC = 21 × 102 × θ = 50θ
1
Area of sector BOD = 2 × 62 × θ = 18θ
Area of shaded region = 50θ – 18θ = 32θ
32θ = 25.6
b)
θ = 25.6 ÷ 32
θ = 0.8 radians
c) Perimeter of the shaded region
= length of arc AC + length of arc BD + AB + CD
= (10 × 0.8) + (6 × 0.8) + 8
= 20.8 cm