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In the below figure, a square $ O A B C $ is inscribed in a quadrant $ O P B Q $ of a circle. If $ O A=21 \mathrm{~cm} $, find the area of the shaded region."
Given:
A square \( O A B C \) is inscribed in a quadrant \( O P B Q \) of a circle.
\( O A=21 \mathrm{~cm} \).
To do:
We have to find the area of the shaded region.
Solution:
From the figure,
$OABC$ is a square.
$OA = 21\ cm$
Join $OB$,
This implies,
Diagonal of the square $\mathrm{OB}=\sqrt{2} \times \mathrm{OA}$
$=\sqrt{2} \times 21 \mathrm{~cm}$
$=21 \sqrt{2} \mathrm{~cm}$
Radius of the quadrant $=21 \sqrt{2} \mathrm{~cm}$
Therefore,
Area of the shaded region $=$ Area of quadrant $-$ Area of square
$=\frac{1}{4} \pi r^{2}-(\mathrm{OA})^{2}$
$=\frac{1}{4} \times \frac{22}{7} \times(21 \sqrt{2})^{2}-(21)^{2}$
$=\frac{11}{14} \times 441 \times 2-441$
$=441(\frac{22}{14}-1)$
$=441 \times \frac{8}{14}$
$=252 \mathrm{~cm}^{2}$
The area of the shaded region is $252\ cm^2$.