Find the required area

Geometry Level 2

A rectangle whose area is 32 and perimeter is 24 is inscribed in a circle as shown. Find the area of the shaded region.

π 2 ( 80 ) − 32 \dfrac{\pi}{2}(80) - 32 20 π − 31 20\pi - 31 32 π − 20 32\pi - 20 4 ( 5 π − 8 ) 4(5\pi - 8)

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1 solution

Naren Bhandari
Feb 12, 2018

Drawing A B AB diagonal of the inscribed rectangle be constructed. Area of rectangle A R = l × b = 32 A_R = l\times b = 32 and perimeters rectangle P R = 2 ( l + b ) = 24 ⟹ l + b = 12 P_R = 2(l+b) =24 \implies l+b =12

Area of shaded portion = Area of circle - Area of rectangle

A s = π r 2 − l × b = π ( A B ) 2 4 − 32 = π l 2 + b 2 4 − 32 = π ( l + b ) 2 − 2 l b 4 − 32 = π 144 − 64 4 − 32 A s = 5 ( 4 π − 8 ) \begin{aligned} & A_s = \pi r^2 - l\times b \\& = \frac{\pi (AB)^2}{4}- 32\\& = \pi \frac{l^2+b^2}{4} - 32 \\& = \pi\frac{(l+b)^2-2lb}{4} - 32 \\& = \pi\frac{144-64}{4} -32 \\& A_s = 5(4\pi -8)\end{aligned}

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