Question

the circumference of a circle is equal to the perimeter of a square, then their areas are in the ratio ?

Answer 1

Answer:

Step-by-step explanation

2πr = 4 x side

πr = 2 x side

Side = πr/2

Ratio of circumference and perimeter

2πr/4s (s= side)

πr/s

πr/πr/2

2πr/πr

2/1

The ratio is 2:1

Answer 2

Circumference of circle = $2\pi r$

Perimeter of square = $4a$

It is given that both are equal. So,

$2 \pi r = 4a$

$\pi r = 2a$

$r = \dfrac{2a}{\pi}$

Hence, we have found out the relation between the radius of the circle and the side of square.

Now, we have to find the ratio of areas:

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

We know that $r = \dfrac{2a}{\pi}$. So, let's substitute this in the formula:

$= \dfrac{\pi \left(\dfrac{2a}{\pi} \right)^{2}}{a^{2}}$

$= \dfrac{\left(\dfrac{\pi 4a^{2}}{\pi^{2}}\right)}{a^{2}}$

The $\pi$ and $\pi^{2}$ can be cancelled as follows:

$= \dfrac{\left(\dfrac{4a^{2}}{\pi}\right)}{a^{2}}$

The $a^{2}$ can be cancelled out.

$= \dfrac{\left(\dfrac{4}{\pi}\right)}{1}$

$= \dfrac{4}{\pi}$

Therefore, the required ratio is $4:\pi$.