Question

In figure a square $OABC$ is inscribed in a quadrant $OPBQ$ of a circle. If $OA=21cm$, find the area of the shaded region.

Answer 1

$OB$ is the diagonal of square $OABC$

$⟹OB=\sqrt{2}\times OA$ $[\because$ length of diagonal of any square is $\sqrt{2}\times$ side $]$

$⟹OA=\frac{OB}{\sqrt{2}}$

$⟹OA=\frac{r}{\sqrt{2}}$ $[\because OB=r$ (radius of the given quadrant $]$

$⟹O{A}^2=\frac{r^2}{2}$

Area of the shaded region $=$ Area of $OPBQ$ $-$ Area of $OABC$

$\therefore Area=\frac{\pi r^2}{4}-\frac{r^2}{2}$

$=\frac{22\times 21\times 21}{7\times 4}-\frac{441}{2}$

$=\frac{693}{2}-\frac{441}{2}$

$=\frac{252}{2}$

$=126{\text{cm}}^2$