Circles and squares are two common geometric shapes encountered very often in our daily life. In this series we present a class of fun problems involving circles in a square.
Look at the following figure, a quarter circle is drawn inside a unit square.
Let $[XY…Z]$ denote the area of the region with vertices of $X$, $Y$, …, $Z$. The area of the region bounded by line $\overline{AB}$, $\overline{AD}$ and arc $\stackrel\frown{BD}$ is $$[ABD] = [ABCD]\ -\ [BCD] = 1\ -\ \frac{\pi}{4}$$
In the following figure, another quarter circle is drawn: 
The area of the region bounded by two arcs with end points as $B$ and $D$ is: $$[BDB] = 1\ -\ 2 \cdot [ABD] = 1\ -\ 2 \cdot (1\ -\ \frac{\pi}{4}) = \frac{\pi}{2}\ -\ 1$$
To be continued…