**Question:**

If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.

**Solution:**

If a square is inscribed in a circle, then the diagonals of the square are diameters of the circle.

Let the diagonal of the square be *d* cm.

Thus, we have:

Radius, $r=\frac{d}{2} \mathrm{~cm}$

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

$=\pi \frac{d^{2}}{4} \mathrm{~cm}^{2}$

We know :

$d=\sqrt{2} \times$ Side

$\Rightarrow$ Side $=\frac{d}{\sqrt{2}} \mathrm{~cm}$

Area of the square $=(\text { Side })^{2}$

$=\left(\frac{d}{\sqrt{2}}\right)^{2}$

$=\frac{d^{2}}{2} \mathrm{~cm}^{2}$

Ratio of the area of the circle to that of the square:

$=\frac{\pi \frac{d^{2}}{4}}{\frac{d^{2}}{2}}$

$=\frac{\pi}{2}$

Thus, the ratio of the area of the circle to that of the square is $\pi: 2$.