Question

In the given figure, if the length of the diagonal of a square which is inscribed in the circle is $5\text{cm}$, then the area of the shaded region is

• A. $25\text{sq. cm}$
• B. $\frac{25}{14}\text{sq. cm}$
• C. $\frac{25}{2}\text{sq. cm}$
• D. $\frac{25}{16}\text{sq. cm}$

Answer 1

The correct option is B $\frac{25}{14}\text{sq. cm}$

$ABCD$ is a square.


Length of diagonal, $AC=5\text{cm}$
Since, diagonal of a square is the diameter of the circle.
$\therefore$ Diameter of the circle $=5\text{cm}$
$\Rightarrow$ Radius of the circle $\left(r\right)=\frac{5}{2}\text{cm}$
Area of minor sector $AOB=\frac{\theta }{{360}^{\circ }}\times \pi r^2$
$=\frac{{90}^{\circ }}{{360}^{\circ }}\times \frac{22}{7}\times \frac{5}{2}\times \frac{5}{2}$
(Diagonals of a square bisect each other at ${90}^{\circ }$)
$=\frac{550}{16\times 7}{\text{cm}}^2$

Area of $\Delta AOB=\frac{1}{2}\times OA\times OB$
$=\frac{1}{2}\times \frac{5}{2}\times \frac{5}{2}$ $(\because OA=OB=\text{radius})$
$=\frac{25}{8}{\text{cm}}^2$
$\Rightarrow$ Area of shaded region $=$ Area of minor sector $AOB-$ Area of $\Delta AOB$

$=\frac{550}{16\times 7}-\frac{25}{8}$

$=\frac{550-25\times 14}{16\times 7}$

$=\frac{200}{16\times 7}=\frac{25}{14}{\text{cm}}^2$

Hence, the correct answer is option b.