Question

In figure, arcs have been drawn with radii 14 cm each and with centres P, Q and R. Find the area of the shaded region.

Answer 1

Given that, radii of each arc (r) = 14 cm

Now, area of the sector with central ∠P = `(∠P)/360^circ xx pir^2`

= `(∠P)/360^circ xx pi xx (14)^2` cm²    ......[∵ Area of any sector with central angle θ and radius r = `(pir^2)/360^circ xx theta`]

Area of the sector with central angle = `(∠Q)/360^circ xx pir^2 = (∠Q)/360^circ xx pi xx (14)^2  cm^2`

And area of the sector with central-angle R = `(∠R)/360^circ xx pir^2 xx (∠R)/360^circ xx  pi xx (14)^2  cm^2`

Therefore, sum of the areas (in cm²) of three sectors

= `(∠P)/360^circ xx pi xx (14)^2 + (∠theta)/360^circ xx pi xx (14)^2 + (∠R)/360^circ xx pi xx (14)^2`

= `(∠P + ∠Q + ∠R)/360^circ xx 196 xx pi`

= `180^circ/360^circ xx 196 pi  cm^2`  .....[Since, sum of all interior angles in any triangle is 180°]

= `98 pi  cm^2 = 98 xx 22/7`

= `14 xx 22`

= 308 cm²

Hence, the required area of the shaded region is 308 cm².