Question

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

• A. $\frac{\pi }{\sqrt{2}}$
• B. $\frac{\pi }{2}$
• C. $\frac{\sqrt{\pi }}{\sqrt{2}}$
• D. $\frac{\sqrt{\pi }}{2}$

Answer 1

The correct option is D $\frac{\sqrt{\pi }}{2}$

Let a be the side of the square and R be the radius of the circle.
Then, $a^2=\pi R^2$
$\frac{R^2}{a^2}=\frac{1}{\pi }$
$\frac{R}{a}=\frac{1}{\sqrt{\pi }}$
Ratio of perimeters of circle and square is
$=\frac{2\pi R}{4a}$
$=\frac{R}{2}\times \frac{\pi }{a}$
$=\frac{\pi }{2}\times \frac{1}{\sqrt{\pi }}$
$=\frac{\sqrt{\pi }}{2}$