If the area of a square is same as the area of a circle, then the ratio of their perimeters, in terms of π, is
(a) $\pi: \sqrt{3}$
(b) $2: \sqrt{\pi}$
(c) $3: \pi$
(d) $\pi: \sqrt{2}$
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).
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