Question

In figure, arcs have been drawn of radius 21 cm each with vertices A, B, C and D of quadrilateral ABCD as centres. Find the area of the shaded region.

Answer 1

Given that, radius of each arc (r) = 21 cm

Area of sector with ∠A

= `(∠"A")/360^circ xx π"r"^2`

= `(∠"A")/360^circ xx π xx (21)^2 "cm"^2`

Area of sector with ∠B

= `(∠"B")/360^circ xx π"r"^2`

= `(∠"B")/360^circ xx π xx (21)^2 "cm"^2`

Area of sector with ∠C

= `(∠"C")/360^circ xx π"r"^2`

= `(∠"C")/360^circ xx π xx (21)^2 "cm"^2`

And area of sector with ∠D

= `(∠"D")/360^circ xx π"r"^2`

= `(∠"D")/360^circ xx π xx (21)^2 "cm"^2`

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

= `(∠"A")/360^circ xx π xx (21)^2 + (∠"B")/360^circ xx π xx (21)^2 + (∠"C")/360^circ xx π xx (21)^2 + (∠"D")/360^circ xx π xx (21)^2`

= `π/360^circ xx (21)^2 xx [∠"A" + ∠"B" + ∠"C" + ∠"D"]`

= `π/360^circ xx (21)^2 xx 360^circ`  ...[∵ Sum of all interior angles in a quadrilateral = 360°]

= `22/7 xx 21 xx 21`

= 22 × 3 × 21

= 1386

Hence, the required area of the shaded region is 1386 cm².