21. The shaded design shown in the diagram is made by drawing eight circular arcs, all with the same radius. The centres of four arcs are the vertices of the square; the centres of the four touching arcs are the midpoints of the sides of the square. The diagonals of the square have length 1 .
What is the total length of the border of the shaded design?
A $2 \pi$
B $\frac{5}{2} \pi$
C $3 \pi$
D $\frac{7}{2} \pi$
E $4 \pi$
OLUTION
The arcs whose centres are the vertices of the square are semicircular arcs, and the arcs whose centres are the midpoints of the sides of the square form three-quarter circles. Let all these arcs have radius $r$. Then the length of the border is
$$
4 \times\left(\frac{1}{2} \times 2 \pi r\right)+4 \times\left(\frac{3}{4} \times 2 \pi r\right)=10 \pi r .
$$

Consider the line $P Q$ joining the midpoints of two adjacent sides of the square as shown. Clearly the length of $P Q$ is half of the length of the diagonal of the square. So $P Q$ has length $\frac{1}{2}$. $P Q$ joins the midpoints of two touching arcs and so its length is twice the radius of these arcs. Hence $r=\frac{1}{4}$. Therefore the length of the border of the shaded design is $10 \pi \times \frac{1}{4}=\frac{5}{2} \pi$.