18. The circumference of a circle with radius 1 is divided into four equal arcs. Two of the arcs are 'turned over' as shown.
What is the area of the shaded region?
A 1
B $\sqrt{2}$
C $\frac{1}{2} \pi$
D $\sqrt{3}$
E 2
Solution
$\mathbf{E}$

Because the four arcs are equal, their endpoints form a square. This square is shown in the diagram.

The four segments $P, Q, R$ and $S$, indicated in the diagram, are congruent. The shaded region is made up of the region $T$, together with the segments $P$ and $Q$. It follows that the area of this region is equal to that of the square which is made up of $T$ and the two segments $R$ and $S$.

The diagonals of the square are diameters of the circle. Therefore they have length 2 . It follows that the square has sides of length $\sqrt{2}$. Hence the area of the square is $\sqrt{2} \times \sqrt{2}$, that is, 2 .

Therefore the area of the shaded region is 2 .