Question

Find the area of the segment AYB shown in the figure, if the radius of the circle is 21 cm and ∠AOB = 120°. `["Use"  π = 22/7]`

Answer 1

Area of segment AYB = Area of sector – Area of triangle   ...(i)

Area of sector = `θ/360^circ xx πr^2`

= `120^circ/360^circ xx 22/7 xx 21 xx 21`

= `1/3 xx 22/7 xx 21 xx 21`

= 22 × 21

= 462 cm²

Area of triangle OAB = `r^2 sin^2  θ/2 cos  θ/2`

= `21 xx 21 xx sin  120^circ/2 cos  120^circ/2`

= 441 × sin 60° × cos 60°

= `441 xx sqrt(3)/2 xx 1/2`

= `(441sqrt(3))/4 cm^2`

From equation (i),

Area of segment AYB = `(462 - (441sqrt(3))/4) cm^2`