Question

Find the area of the shaded regions below. Give your answer as a completely simplified exact value in terms of π (no approximations).

Answer 1

The area of the shaded region is 36π - 72 cm²

Explanation:

Given

The radius of the circle segment, r = 12cm

The angle subtended at the centre of the circle is right angled, so

θ = 90°

Convert to radians

θ = 90π/180

θ = ½π

The shaded region is a segment;

To calculate the area of the segment, we make use of the following formula;

A = ½r²(πθ/180 - sinθ) in radians

By substituton,

A = ½ * 12² * (π/2 - sin90)

A = ½ * 144(π/2 - 1)

A = 72(π/2 - 1)

A = 36π - 72 cm²

Area of the shaded region is 36π - 72 cm²

Answer 2

(36π -72) cm²

Explanation:

The area of a segment that subtends arc α (in radians) is given by ...

A = (1/2)r²·(α - sin(α))

Here, you have r = 12 cm and α = π/2 radians, so the area of the segment is ...

A = (1/2)(12 cm)²·(π/2 -1) = (36π -72) cm²