Find the area of the shaded part

Geometry Level 2

The figure above is formed by describing semicircular arcs within the square upon the four sides as diameters. Given that the side length of the square us 8, what is the area of the shaded part?

32 π − 64 32\pi - 64 128 − 29 π 128 - 29\pi 25 π − 42 25\pi - 42 16 π − 14 16\pi -14

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1 solution

Consider Figure 1:

Let X X be the area of the semicircle, and Y Y be the area of the part as shown. Then

X = 1 2 ( π ) ( 4 2 ) = 8 π X=\dfrac{1}{2}(\pi)(4^2)=8\pi

Y = a r e a o f t h e s q u a r e − 2 X 2 = 8 2 − 2 ( 8 π ) 2 = 32 − 8 π Y=\dfrac{area~of~the~square-2X}{2}=\dfrac{8^2-2(8\pi)}{2}=32-8\pi

Consider Figure 2:

a r e a s h a d e d = a r e a o f t h e s q u a r e − 4 Y = 8 2 − 4 ( 32 π − 64 ) = 32 π − 64 area~shaded=area~of~the~square-4Y=8^2-4(32\pi-64)=\large{\color{#69047E}\boxed{32\pi - 64}}

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