If the area of a square is same as the area of a circle,

We have given that area of a circle of radius r is equal to the area of a square of side a.

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

$\therefore a=\sqrt{\pi} r$

We have to find the ratio of the perimeters of circle and square.

$\therefore \frac{\text { Perimeter of circle }}{\text { Perimeter of square }}=\frac{2 \pi r}{4 a}$............(1)

Now we will substitute $a=\sqrt{\pi} r$ in equation (1).

$\frac{\text { Perimeter of circle }}{\text { Perimeter of square }}=\frac{2 \pi r}{4 \sqrt{\pi} r}$

$\therefore \frac{\text { Perimeter of circle }}{\text { Perimeter of square }}=\frac{\pi}{2 \sqrt{\pi}}$

$\therefore \frac{\text { Perimeter of circle }}{\text { Perimeter of square }}=\frac{\sqrt{\pi}}{2}$

Therefore, ratio of their perimeters is $\sqrt{\pi}: 2$.

Hence, the correct answer is (b).