Question

The area of a square is the same as the area of a square. Their perimeters are in the ratio
(a) 1 : 1
(b) 2 : π
(c) π : 2
(d) $\sqrt{\mathrm{\pi }}:2$

Answer 1

(d) $\sqrt{\mathrm{\pi }}:2$
Let a be the side of the square.
We know:
Area of a square$=a^2$
Let r be the radius of the circle.
We know:
Area of a circle$=\mathrm{\pi }{r}^2$
Because the area of the square is the same as the area of the circle, we have:
$$\begin{gathered} a^2=\mathrm{\pi }{r}^2 \\ \Rightarrow \frac{r^2}{a^2}=\frac{1}{\mathrm{\pi }} \\ \Rightarrow \frac{r}{a}=\frac{1}{\sqrt{\mathrm{\pi }}} \end{gathered}$$
∴ Ratio of their perimeters$=\frac{2\mathrm{\pi }r}{4a}\left[\mathrm{Because}\mathrm{perimeter}\mathrm{of}\mathrm{the}\mathrm{circle}\mathrm{is}2\mathrm{\pi }r\mathit{}\mathrm{and}\mathrm{perimeter}\mathrm{of}\mathrm{the}\mathrm{square}\mathrm{is}4a\right]$

$$\begin{gathered} =\frac{\mathrm{\pi }r}{2a} \\ =\frac{\mathrm{\pi }}{2}\times \frac{r}{a} \\ =\frac{\mathrm{\pi }}{2}\times \frac{1}{\sqrt{\mathrm{\pi }}}\left[\mathrm{Since}\frac{r}{a}=\frac{1}{\sqrt{\mathrm{\pi }}}\right] \\ =\frac{\sqrt{\mathrm{\pi }}}{2} \\ =\sqrt{\mathrm{\pi }}:2 \end{gathered}$$