Question

AB and CD are diameters of a circle with centre O and radius 7 cm. If ∠BOD = 30°, then find the area and perimeter of the shaded region.

Answer 1

Given radius = 7 cm

∠BOD = 30°

Area of sector BOD = `θ/(360°) πr^2`

= `30/(360°) xx π(7)^2`

= `1/12 π xx 49`

= `(49π)/12` cm²

Length of arc BD = `θ/(360°) 2πr`

= `(30°)/(360°) xx 2π(7)`

= `1/12 xx 14π`

= `(7π)/6` cm

Area of sector AOC = `(30°)/(360°) π(7)^2`   ...(∵ ∠BOD = ∠AOC = 30°)

 = `1/12 π xx 49`

= `(49π)/12` cm²

Length of arc AC = `(30°)/(360°) xx 2π(7)`

= `1/12 xx 14π`

= `(7π)/6` cm

Total area of shaded region = Area of sector AOC + Area of sector BOD

= `(49π)/12 + (49π)/12`

= `(98π)/12`

= `98/12 xx 22/7`

= `(14 xx 11)/6`

= `154/6`

= 25.67 cm²

Perimeter of shaded region = OD + OB + Length of arc

= BD + OA + OC + Length of arc AC   ...[∵ OD = OB = OA = OC = 7 cm]

= `7 + 7 + (7π)/6 + 7 + 7 + (7π)/6`

= `(14π)/6` + 28 cm

= `14/6 xx 22/7 + 28`

= `22/3 + 28`

= `(22 + 84)/3`

= `106/3`

= 35.33 cm