Question

In the given figure, chord AB subtends an angle of 120° at the centre of the circle with radius 7 cm. Find (i) perimeter of major sector OACB, and (ii) area of the shaded segment, if area of Δ OAB = 21.2 cm².

Answer 1

Given: Radius = 7 cm

Angle subtended at the centre θ = 120°

Area of Δ OAB = 21.2 cm²

(i) Perimeter of Major Sector OACB

= 360° − 120°

= 240°

Length of major arc ACB

Arc length = `θ/(360°) xx 2πr`

= `240/360 xx 2pi xx 7`

= `2/3 xx 14pi`

= `(28pi)/3`

=`28/3 xx 22/7    ...(∵ pi = 22/7)` 

= `616/21`

= 29.33 cm

Perimeter of major sector:

Perimeter = Major arc length + 2r

= 29.33 + 7 + 7

= 29.33 + 14

= 43.33 cm

(ii) Area of the Shaded Segment

Area of minor sector OAB

Area = `120/360 xx πr^2`

= `1/3 xx pi xx 7^2`

= `1/3 xx 22/7 xx 49    ...(∵ pi = 22/7)` 

= `(22 xx 49)/(3 xx 7)`

= `1078/21`

= 51.33 cm²

Area of shaded segment

= Area of minor sector OAB − Area of triangle OAB

= 51.33 − 21.2

= 30.13 cm²