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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.
Solution
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.
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