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The figure shows the rectangle ABCD with a semicircle and a circle inscribed inside in it as shown. What is the ratio of the area of the circle to that of the semicircle?
Let the center be O and the point at which the semicircle intersects CD be P.
Let the radius of the semicircle be R and the circle be r.
OP = R and OC = R$$\sqrt{2}$$
OC - OT = CC' - TC'
$$R\sqrt{2} - R - 2r$$ = $$r\sqrt{2} - r$$
=> $$R\sqrt{2} - R$$ = $$r\sqrt{2} + r$$
=> r = $$\frac{(\sqrt{2}-1)R}{\sqrt{2}+1}$$
=> r = $$(\sqrt{2}-1)^2$$R
Ratio of areas will be $$r^2 : \frac{R^2}{2}$$ = $$2(\sqrt{2}-1)^4$$ : 1
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