In Figure 9, a square $OPQR$ is inscribed in a quadrant $OAQB$ of a circle. If the radius of the circle is $6\sqrt{2}$ cm, find the area of the shaded region.

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Question

In Figure 9, a square $OPQR$ is inscribed in a quadrant $OAQB$ of a circle. If the radius of the circle is $6\sqrt{2}$ cm, find the area of the shaded region.

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Accepted Answer

Step 1: Let side of square be $a$ cm. Diagonal of square $= a\sqrt{2}$ = radius $= 6\sqrt{2}$ cm. So $a = 6$ cm.
Step 2: Area of shaded region $=$ Area of quadrant $-$ Area of square $= \frac{90}{360} 	imes \pi r^2 - a^2 = \frac{1}{4} 	imes \frac{22}{7} 	imes (6\sqrt{2})^2 - 36$
Step 3: $= \frac{22 	imes 72}{28} - 36 = \frac{396 - 252}{7} = \frac{144}{7} = 20.57$ cm$^2$

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