triangle in a half of a circle

Geometry Level pending

A triangle with the maximum area is inscribed in a semi-circle with radius of 2 as shown. Find the area of the shaded part.

Note: The figure is not drawn to scale.

− 2 π + 4 -2\pi+4 π − 5 \pi-\sqrt{5} 2 π − 4 π \sqrt{2\pi}-4\pi 2 π − 4 2\pi-4

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1 solution

For maximum area, the height must be equal to the radius of the circle.

shaded area = area of semicircle - area of triangle = 1 2 π ( 2 2 ) − 1 2 ( 4 ) ( 2 ) = 2 π − 4 \text{shaded area = area of semicircle - area of triangle}=\dfrac{1}{2} \pi (2^2)-\dfrac{1}{2}(4)(2)=\boxed{2\pi -4}

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