Question

In the figure, a square of diagonal $ 8 \mathrm{cm}$ is inscribed in a circle. Find the area of the shaded region.

Answer 1

Step 1: Find the area of the square.

Given: Diagonal of the square $=8\mathrm{cm}$

Diameter of the circle $=$Diagonal of the square $=8\mathrm{cm}$

Let the side of the square be $a$.

We know that side of the square and diagonal follow the relation given below:

$\mathrm{a}=\sqrt{\frac{{\mathrm{d}}^2}{2}}$ ($a$ is the side of the square and $d$ is diagonal)

Therefore,

$\mathrm{a}=\sqrt{\frac{8^2}{2}}=\sqrt{\frac{64}{2}}=\sqrt{32}\mathrm{cm}$

Area of square

$$\begin{gathered} ={\mathrm{a}}^2 \\ ={\left(\sqrt{32}\right)}^2 \\ =32{\mathrm{cm}}^2 \end{gathered}$$

Step 2: Find the area of the circle.

Area of circle$={\mathrm{\pi r}}^2$

$$\begin{gathered} =\frac{22}{7}\times {\left(4\right)}^2\left(\mathrm{radius}=\frac{\mathrm{diameter}}{2}\right) \\ =50.28{\mathrm{cm}}^2 \end{gathered}$$

Step 3: Find the area of the shaded region.

Area of the shaded region $=$ Area of the circle$-$ Area of the square

$$\begin{gathered} =50.28-32 \\ =18.28{\mathrm{cm}}^2 \end{gathered}$$

Hence the area of the shaded region is $18.28{\mathrm{cm}}^2$.