Question

In Figure 2, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region.

Answer 1

Radius of inner circle, OC = 21 cm

Radius of outer circle, OA = 42 cm

Area of a circle with radius R = `piR^2 = pi(42)^2`

Area of a circle with radius r = `pir^2 = pi(21)^2`

Area of sector AOB = `theta/360 xx piR^2 = 60/360 xx pi(42)^2 = (pi(42)^2)/6`

Area of sector COD = `theta/360 xx pir^2 = 60/360 xx pi(21)^2 = (pi(21)^2)/6`

Area of shaded portion = Area of circle with radius R - Area of circle with radius r - [Area of sector AOB - Area of sector COD]

= `pi(42)^2 - pi(21)^2 - [(pi(42)^2)/6 - (pi(21)^2)/6]`

= `pi[(42)^2 - (21)^2 - 1/6[(42)^2 - (21)^2]]`

= `pi[((42)^2 - (21)^2)(1 - 1/6)]`

= `pi[(42 - 21)(42 + 21)5/6]`

= `22/7 xx 5/6 xx 21 xx 63`

= 3465 cm²