Question

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

Answer 1

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

$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{circle}=\mathrm{\pi }{r}^2$
= $\mathrm{\pi }\frac{d^2}{4}{\mathrm{cm}}^2$

$$\begin{gathered} \mathrm{We}\mathrm{know}: \\ d=\sqrt{2}\times \mathrm{Side} \\ \Rightarrow \mathrm{Side}=\frac{d}{\sqrt{2}}\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{square}=(\mathrm{Side}{)}^2 \\ ={\left(\frac{d}{\sqrt{2}}\right)}^2 \\ =\frac{d^{\mathit{2}}}{2}{\mathrm{cm}}^2 \end{gathered}$$

Ratio of the area of the circle to that of the square:
$$\begin{gathered} =\frac{\pi {\displaystyle \frac{d^2}{4}}}{{\displaystyle \frac{d^2}{2}}} \\ =\frac{\mathrm{\pi }}{2} \end{gathered}$$
Thus, the ratio of the area of the circle to that of the square is $\mathrm{\pi }:2$.