Question

In the figure, a square inscribed in a circle with centre O and radius $r$. The area of the region which is not included in the square is ______

• A. $r^2(\pi -2)$
• B. $2r(2-\pi )$
• C. $\pi (r^2-2)$
• D. ${\pi }^2-8r$

Answer 1

Answer: (A)

$r^2(\pi -2)$

Area of shaded region $=$ Area of circle $-$ Area of square
But, diameter of circle of radius $r$ $=$ diagonal of square of each side $l$
$\Rightarrow 2r=\sqrt{2}l$

$\Rightarrow r=\frac{l}{\sqrt{2}}$

$\Rightarrow l=r\sqrt{2}$
Area of shaded Region $=r^2-l^2$

$=\pi r^2-(r\sqrt{2}{)}^2$
$=\pi r^2-2r^2$

$=r^2(\pi -2)$ sq units