If the perimeter of a circle is equal to that of a square, then find the ratio of their areas.

Given: The perimeter of a circle is equal to that of a square.

To do: To find the ratio of their areas.

Solution:

As given the perimeter of the circle is equal to that of the square.

$P_{circle}=P_{square}$

Let $r$ be the radius of the circle & $a$ be the side of square, then

$2\pi r=4a$

$\frac{r}{a}=\frac{4}{2\pi}=\frac{2}{\pi}$

Now, $\frac{Area\ of\ the\ circle}{Area\ of\ the\ square}=\frac{\pi r^2}{a^2}$

$\Rightarrow \frac{Area\ of\ the\ circle}{Area\ of\ the\ square}=\frac{\pi\times2^2}{\pi^2}$

$=\frac{4}{\pi}$

Hence the ratio of Area of a circle to that of a square is $\frac{4}{\pi}$.

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Updated on: 10-Oct-2022

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