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A square of diagonal 8 cm is inscribed in a circle. Find the area of the region lying inside the circle and outside the square.
Given:
A square of diagonal 8 cm is inscribed in a circle.
To do:
We have to find the area of the region lying inside the circle and outside the square.
Solution:
Let the side of the square be $a$ and the radius of the circle be $r$.
The length of the diagonal of the square $= 8\ cm$
Diagonal of the square of side $a=\sqrt{2} a$
This implies
$\sqrt{2}a=8$
$\Rightarrow a=4 \sqrt{2} \mathrm{~cm}$
Diagonal of the square $=$ Diameter of the circle
This implies,
Diameter of the circle $=8\ cm$
Radius of the circle $r=\frac{\text { Diameter }}{2}$
$\Rightarrow r=\frac{8}{2}$
$\Rightarrow r=4 \mathrm{~cm}$
Therefore,
Area of the circle $=\pi r^{2}$
$=\pi(4)^{2}$
$=16 \pi \mathrm{cm}^{2}$
Area of the square $=a^{2}$
$=(4 \sqrt{2})^{2}$
$=32 \mathrm{~cm}^{2}$
The area of the region lying inside the circle and outside the square $=$ Area of the circle $-$ Area of the square
$=(16 \pi-32) \mathrm{cm}^{2}$
The area of the region lying inside the circle and outside the square is $(16 \pi-32) \mathrm{cm}^{2}$.