Question

The area of a circle is the same as the area of a square. Their perimeters are in the ratio.

(a) 1 : 1 (b) $\frac{2}{\sqrt{\pi }}$ (c) $\pi$ : 2 (d) $\sqrt{\pi }$ : 2

Answer 1

Given that areas of circle and square are equal.

If a is the side of a square, then the area of the square is $a^2$ units

If r is the radius of a circle, then the area of the circle is $\pi r^2$ units

Thus, from the given condition,

$a^2=\pi r^2$

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

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

The perimeter of a square is 4a

The perimeter of a circle is $2\pi \times r$

Thus the ratio of the perimeter of a square and a circle is

$\frac{4a}{2\pi r}$=$\frac{2r\sqrt{p}i}{\pi r}$ [ $a=r\sqrt{\pi }$ ]

Thus the ratio is $\frac{2}{\sqrt{\pi }}$