Problem is to find the ratio of the area of the circle to that of the semi-circle.
Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right is touching the semi-circle as well as the two sides of the rectangle.
I don't have any idea on where to start. I have seen these things, but this question is completely different. If even I could compute the diagonal $B$ to center of semi-circle, there would be a very tiny gap at the top-right.
Any help/hints on this is highly appreciated.
Let $R$ be the radius of the semicircle, and $r$ the radius of the little circle. Let $O$ be the centre of the semicircle. Then $OB=R\sqrt{2}$ and $EB=r\sqrt{2}$. It follows that $$R\sqrt{2}=R+r+r\sqrt{2}.$$