In the adjoining figure, $$A,B,C,D$$ are the vertices of a square of side $$3$$ units. All the semicircles are equal. The area of the shaded region in square units is
[image]
Solution
Each small semicircle has radius $$\frac{1}{2}$$ and area $$\frac{\pi}{8}$$. The outer boundary adds twelve such semicircles to the square, while the central white region removes a unit square and four such semicircles. Hence the shaded area is $$9+12\left(\frac{\pi}{8}\right)-\left(1+4\left(\frac{\pi}{8}\right)\right)=8+\pi$$.
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