Question

In the figure below, $RSTV$ is a square inscribed in a circle $O$ and radius $r$. What is the total area of shaded region ?

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

Answer 1

The correct option is A $r^2(\pi -2)$

Area of a circle of radius '$r$' $=\pi r^2$
The diagonal of the square $=2r$
But diagonal of a square $=\sqrt{2}\times \text{side}$
$\Rightarrow$ side of the square $=\frac{2r}{\sqrt{2}}=\sqrt{2}r$
Area of shaded region $=$ Area of circle $-$ Area of square

$=\pi r^2-{\text{side}}^2=\pi r^2-{\left(\sqrt{2}r\right)}^2$

$=\pi r^2-2r^2$

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