20. The diagram shows a square $P Q R S$ with edges of length 1 , and four arcs, each of which is a quarter of a circle. Arc $T R U$ has centre $P$; arc $V P W$ has centre $R$; arc $U V$ has centre $S$; and arc $W T$ has centre $Q$.
What is the length of the perimeter of the shaded region?
A 6
B $(2 \sqrt{2}-1) \pi$
C $\left(\sqrt{2}-\frac{1}{2}\right) \pi$
D $2 \pi$
E $(3 \sqrt{2}-2) \pi$
Solution
B

The square $P Q R S$ has side length 1 , and hence its diagonal $P R$ has length $\sqrt{2}$. So the arc $T R U$ with centre $P$ has radius $\sqrt{2}$. The arc is a quarter of a circle. Hence its length is $\frac{1}{4} \times 2 \pi \sqrt{2}$, that is $\frac{1}{2} \sqrt{2} \pi$.
Similarly, the length of the arc $V P W$ is $\frac{1}{2} \sqrt{2} \pi$.
The radius of the arc $U V$ is equal to the length of $S U$. Because $P U$ has length $\sqrt{2}$ and $P S$ has length 1 , the length of $S U$ is $\sqrt{2}-1$. The arc $U V$ is one quarter of a circle with this radius. Hence the length of this arc is $\frac{1}{4} \times 2 \pi(\sqrt{2}-1)$, that is $\frac{1}{2}(\sqrt{2}-1) \pi$.
Similarly, the arc $W T$ has length $\frac{1}{2}(\sqrt{2}-1) \pi$.
Therefore the total length of the perimeter of the shaded region is given by
$$
\frac{1}{2} \sqrt{2} \pi+\frac{1}{2} \sqrt{2} \pi+\frac{1}{2}(\sqrt{2}-1) \pi+\frac{1}{2}(\sqrt{2}-1) \pi=(2 \sqrt{2}-1) \pi .
$$