# Two overlapping circles with equal radius form a shaded region as shown in the figure. Express the area of the region and the complete perimeter (combined arc length) in terms of r and the distance between center, #D#? Let #r=4 and D=6# and calculate?

see explanation.

Given

Given

Area GEF (red area)

Yellow Area

arc perimeter

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The area of the shaded region is (2\pi r^2 - 2\left(\frac{1}{2} \pi r^2\right) = \pi r^2). The perimeter of the shaded region is (2\pi r + 2\pi \left(r\cos^{-1}\left(\frac{D}{2r}\right)\right)).

Given (r=4) and (D=6),

The area of the shaded region is (\pi \times 4^2 = 16\pi) square units.

The perimeter of the shaded region is (2\pi \times 4 + 2\pi \left(4\cos^{-1}\left(\frac{6}{2\times 4}\right)\right) = 8\pi + 2\pi \left(4\cos^{-1}\left(\frac{3}{4}\right)\right)). Using trigonometric identities, (\cos^{-1}\left(\frac{3}{4}\right) = \frac{\pi}{5}), So, the perimeter is (8\pi + 2\pi \times 4 \times \frac{\pi}{5} = 8\pi + \frac{8\pi}{5} = \frac{48\pi}{5}).

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