21. The perimeter of a logo is created from two vertical straight edges, two small semicircles with horizontal diameters and two large semicircles. Both of the straight edges and the diameters of the small semicircles have length 2 . The logo has rotational symmetry as shown.
What is the shaded area?
A 4
B $4-\pi$
C 8
D $4+\pi$
E 12
SolUTION
D

The diagram on the right shows the top half of the shaded region.
This region is made up of the right-angled triangle $P Q R$ and the semicircle with diameter $P R$, but with the semicircle with $Q R$ as diameter removed.
The area of the triangle $P Q R$ is $\frac{1}{2}(P Q \times Q R)=\frac{1}{2}(2 \times 2)=2$.
By Pythagoras' Theorem, applied to the right-angled triangle $P Q R$, we have $P R^{2}=2^{2}+2^{2}=8$. Therefore $P R=2 \sqrt{2}$.
Hence the semicircle with diameter $P R$ has radius $\sqrt{2}$. Hence the area of this semicircle is $\frac{1}{2}\left(\pi(\sqrt{2})^{2}\right)=\pi$.
The semicircle with $Q R$ as diameter has radius 1 , and therefore its area is $\frac{1}{2}\left(\pi 1^{2}\right)=\frac{1}{2} \pi$.
It follows that the area of the shaded region in the diagram above is
$$
2+\pi-\frac{1}{2} \pi=2+\frac{1}{2} \pi \text {. }
$$

This area is half of the shaded area in the logo.
Therefore the shaded area in the logo is $2 \times\left(2+\frac{1}{2} \pi\right)=4+\pi$.
FOR INVESTIGATION
21.1 Find the length of the perimeter of the shaded area.