In the adjoining diagram, $$ABCD$$ is a square. $$P,Q,R,S$$ are the mid points of sides $$AB,BC,CD$$ and $$DA$$ respectively. $$PQ$$ and $$SR$$ are quadrants of circles with $$B,D$$ as the centres and $$BP$$ as radius. $$PS$$ and $$RQ$$ are part of the circle passing through $$P,Q,R$$ and $$S$$. If $$AB=20\text{ cm}$$, the area of the shaded region, in $$\text{cm}^2$$, is

Since $$AB=20\text{ cm}$$, the four midpoints lie on a circle of radius $$10\text{ cm}$$ centered at the centre of the square. The area inside this circle is $$100\pi$$. The two quadrant regions removed from the shaded part have radius $$10\text{ cm}$$ and together have area $$2\left(\frac14\pi\cdot10^2\right)=50\pi$$. Hence the shaded area is $$100\pi-50\pi=50\pi\text{ cm}^2$$.