Question

If the perimeter of a square is equal to the circumference of a circle, then the ratio of their areas $\left(\mathrm{when}\mathrm{\pi }=\frac{22}{7}\right)$ is.
(a) 14 : 11
(b) 11 : 14
(c) 22 : 7
(d) 7 : 22

Answer 1

(b) 11:14
Let P be the perimeter of the square.
Now,
Each side of the square$=\frac{P}{4}$
Let r be the radius of the circle.
We know:
Circumference of the circle$=2\mathrm{\pi }r$
Now,
$$\begin{gathered} 2\mathrm{\pi }r=P \\ \Rightarrow r=\frac{P}{4\mathrm{\pi }} \end{gathered}$$
∴ Area of the square$={\left(\frac{P}{4}\right)}^2$
$=\frac{P^2}{16}$

Also,
Area of the circle $=\mathrm{\pi }{r}^2$
$$\begin{gathered} =\mathrm{\pi }\times {\left(\frac{P}{2\mathrm{\pi }}\right)}^2 \\ =\mathrm{\pi }\times \frac{P^{\mathit{2}}}{4{\mathrm{\pi }}^2} \\ =\frac{P^{\mathit{2}}}{4\mathrm{\pi }} \end{gathered}$$
∴ Required ratio$=\frac{P^{\mathit{2}}}{16}\times \frac{4\mathrm{\pi }}{P^{\mathit{2}}}$
$$\begin{gathered} =\frac{\mathrm{\pi }}{4} \\ =\left(\frac{22}{7\times 4}\right) \\ =\frac{11}{14} \end{gathered}$$

=11:14