21. The diagram shows a shaded semicircle of diameter 4 , from which a smaller semicircle has been removed. The two semicircles touch at exactly three points. What fraction of the larger semicircle is shaded?

A $\frac{2}{\pi}$
B $\frac{1}{2}$
C $\frac{\sqrt{2}}{3}$
D $\frac{\sqrt{2}}{2}$
E $\frac{3}{4 \pi}$
21. B First note that the radius of the larger semicircle is 2 .
So its area is $\frac{1}{2} \times \pi \times 2^{2}=2 \pi$.
Let the radius of the smaller semicircle be $r$.
Then, by Pythagoras' Theorem, $r^{2}+r^{2}=2^{2}$. So $r^{2}=2$.
Therefore the area of the smaller semicircle is $\frac{1}{2} \times \pi \times 2=\pi$.
So the shaded area is $2 \pi-\pi=\pi$.
Hence half the area of the large semicircle is shaded.