The area of a square is the same as the area of a square.
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) $\sqrt{\pi}: 2$

Solution:

(d) $\sqrt{\pi}: 2$

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 $=\pi r^{2}$

Because the area of the square is the same as the area of the circle, we have:

$a^{2}=\pi r^{2}$

$\Rightarrow \frac{r^{2}}{a^{2}}=\frac{1}{\pi}$

$\Rightarrow \frac{r}{a}=\frac{1}{\sqrt{\pi}}$

$\therefore$ Ratio of their perimeters $=\frac{2 \pi r}{4 a} \quad[$ Because perimeter of the circle is $2 \pi r$ and perimeter of the square is $4 a]$

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

$=\frac{\pi}{2} \times \frac{r}{a}$

$=\frac{\pi}{2} \times \frac{1}{\sqrt{\pi}} \quad\left[\right.$ Since $\left.\frac{r}{a}=\frac{1}{\sqrt{\pi}}\right]$

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

$=\sqrt{\pi}: 2$