Question:
In the following figure, OE = 20 cm. In sector OSFT, square OEFG is inscribed. Find the area of the shaded region.
Solution:
We have to find the area of the shaded portion. We have, $\mathrm{OE}=20 \mathrm{~cm}$ and $\mathrm{OEFG}$ is a square.
Use Pythagoras theorem to find OF as,
$\mathrm{OF}=\left(\sqrt{(20)^{2}+(20)^{2}}\right) \mathrm{cm}$
$=20 \sqrt{2} \mathrm{~cm}$
Therefore area of the shaded region,
Area of the shaded region $=($ Area of quadrant of cicle $)-($ Area of square $)$
So,
Area of the shaded region $=\frac{\pi(\mathrm{OF})^{2}}{4}-(\mathrm{OE})^{2}$
$=\left(\frac{22}{7}\right)\left(\frac{(20 \sqrt{2})^{2}}{4}\right)-(20)^{2}$
$=628-400$
$=228 \mathrm{~cm}^{2}$