In the below figure, a square $ O A B C $ is inscribed in a quadrant $ O P B Q $. If $ O A=15 \mathrm{~cm} $, find the area of shaded region (use $ \pi=3.14)."

Given:

A square \( O A B C \) is inscribed in a quadrant \( O P B Q \) of a circle.

\( O A=15 \mathrm{~cm} \).

To do: 

We have to find the area of the shaded region.

Solution:

From the figure,
$OABC$ is a square.

$OA = 15\ cm$

Join $OB$,

This implies,

Diagonal of the square $\mathrm{OB}=\sqrt{2} \times \mathrm{OA}$

$=\sqrt{2} \times 15 \mathrm{~cm}$

$=15 \sqrt{2} \mathrm{~cm}$

Radius of the quadrant $=15 \sqrt{2} \mathrm{~cm}$

Therefore,

Area of the shaded region $=$ Area of quadrant $-$ Area of square

$=\frac{1}{4} \pi r^{2}-(\mathrm{OA})^{2}$

$=\frac{1}{4} \times \frac{22}{7} \times(15 \sqrt{2})^{2}-(15)^{2}$

$=\frac{11}{14} \times 225 \times 2-225$

$=225(\frac{11}{7}-1)$

$=225 \times \frac{11-7}{7}$

$=225\times \frac{4}{7}$

$=128.25 \mathrm{~cm}^{2}$

The area of the shaded region is $128.25\ cm^2$. 

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Updated on: 10-Oct-2022

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