Question

In the adjoining figure, ΔOAB is an equilateral triangle and the area of the shaded region is 750 π cm². Find the perimeter of the shaded region.

Answer 1

1. Identify the central angle

Since ΔOAB is an equilateral triangle, each of its interior angles is 60°. 

Therefore, the central angle of the unshaded minor sector is 60°.

The central angle θ of the shaded major sector is:

θ = 360° – 60°

= 300°

2. Find the radius

The area of a sector is given by the formula:

Area = `θ/360 xx πr^2`

Given the shaded area is 750π cm²:

`750π = 300/360 xx πr^2`

`750 = 5/6 xx r^2`

`r^2 = (750 xx 6)/5`

= 150 × 6

= 900

`r = sqrt(900)`

= 30 cm

3. Calculate the Arc length

The length of the major arc is:

Arc length = `θ/360 xx 2πr`

Arc length = `300/360 xx 2π(30)`

Arc length = `5/6 xx 60π`

Arc length = 50π cm

 4. Find the total perimeter

The perimeter of the shaded region consists of the major arc and the two radii forming the boundary.

Perimeter = Arc length + 2r

Perimeter = 50π + 2(30)

= (50π + 60) cm

Using π ≈ 3.14:

Perimeter = 50(3.14) + 60

= 157 + 60

= 217 cm

The perimeter of the shaded region is (50π + 60) cm, which is approximately 217 cm.