The ratio of the perimeters of a circle and a square of the same area is

Letr be the radius of the circle and a be the side of the square.

Then,

Area of the circle = πr²

Area of the square = a²

πr² = a²

$\Rightarrow \frac{a}{r}=\sqrt{\mathrm{\pi }} ...\left(1\right)$

Perimeter of the circle = 2πr

Perimeter of the square = 4a

⇒ Ratio of the perimeters of the circle and square

= 2πr : 4a

From (1),

$$\begin{gathered} r:a=1:\sqrt{\mathrm{\pi }} \\ \Rightarrow 2\mathrm{\pi r}:4\mathrm{a}=2\mathrm{\pi }:4\sqrt{\mathrm{\pi }}=\sqrt{\mathrm{\pi }}:2 \end{gathered}$$