Question

The following three shapes are based only on squares, semicircles, and quarter circles. Find the perimeter and the area of each shaded part. Give your answer as a completely simplified exact value in terms of π (no approximations).

Answer 1

First compute the area of the square like this: $9^2=81$
There is two semi circles here, so the sum of the two white 
spaces equate the area of a circle. 
the radius of the circle is 9/2. 
So compute the area of the like this:  $\pi (\frac{9}{2})^2$
And the area of the shaded region is the area of the square minus 
the area of the two semi circles, like this:
$81-\pi (\frac{9}{2})^2$
We are done.