Further Trigonometry
Arc and Sector Area

You have already have seen how to find the arc length.

If θ is measured in degrees, then

arc length PQ = × 2πr =

but if θ is measured in radians,

arc length PQ = × 2πr = rθ

The region between the two radii and the arc is called a sector of the circle. There is a simple formula for the area of a sector of a circle subtended by an angle.

Area of sector = × πr2

but if θ is measured in radians,

area of sector = × πr2

area of sector = r2θ

So, using radian measure, the formulae for both arc length and sector area take a simple form, namely

arc length = rθ

sector area = r2θ

You will see how these formulae are used in the following example.

Worked Examples

1
(a)

A silver pendant is made in the form of a sector of a circle as shown opposite. If the radius is 3 cm, what is the angle, θ, in radians so that the area is 6 cm2?

Show me
Sector area= 6 = × 32 × θ
⇒θ = radians
(b)

Another pendant has the same total perimeter, but with radius 2.5 cm. What is the required angle, θ, in radians?

Show me
Total perimeter length= 3 + 3 + 3 ×
= 10 cm

Hence, for the second pendant,

10 = 2.5 + 2.5 + 2.5 × θ

⇒ θ − 2 radians

Exercises

An oil drum of diameter 60 cm is floating as shown in the diagram on the right.

(a)

Given that the arc length PQ is 50 cm, find θ in radians.

radians
(b)

The drum is 1 metre long. Find the volume of the drum which lies below the surface level.

cm3

In the diagram on the right, the length of the arc AB is 20 cm and the area of the sector AOB is 100 cm2.

Form two equations involving r and θ when θ is measured in radians. Solve these equations simultaneously to find r and θ.

r = cm

θ = radians

A stone, swung round on the end of a 200 cm string, completes an arc subtending an angle at the centre of 2 radians every second. Find the speed at which the stone is moving.

ms−1