Question

In the given figure, $ABCD$ is a square of side $21cm$. $AC$ and $BD$ are two diagonals of the square. Two semi circles are drawn with $AD$ and $BC$ as diameters. Find the area of the shaded region (in $c{m}^2$). (Take $\pi =\frac{22}{7}$)

Answer 1

Given : Side $=21cm$,

Let Diagonal of the square $=\sqrt{2}$ (side)

$\therefore$ $AC=BD=21\sqrt{2}$

$\therefore$ $AO=OC=BO=OD=\frac{21\sqrt{2}}{2}$
(Diagonals of square bisect each other)

Area of $△AOD=$ Area of $△BOC=\frac{1}{2}\times \frac{21\sqrt{2}}{2}\times \frac{21\sqrt{2}}{2}=\frac{441}{4}{cm}^2$

Area of semicircle $=\frac{1}{2}\pi r^2$

$=\frac{1}{2}\times \frac{22}{7}\times {\left(\frac{21}{2}\right)}^2=\frac{693}{4}{cm}^2$

Area of shaded region $=$ Area of 2 semicircles $+$ Area of $△AOD+$ Area of $△BOC$

$=2\times \frac{693}{4}+\frac{441}{4}+\frac{441}{4}=\frac{2268}{4}$

$=567{cm}^2$.