Question

The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of π?

Answer 1

Ratio of the area of the square to the area of the circle is $\frac{\pi}{4} : 1$

In terms of fraction $\frac{\pi}{4}$

Solution:

Given that perimeter of a particular square and the circumference of a particular circle are equal

To find:  ratio of the area of the square to the area of the circle

Let the side of square be "a" and radius of circle be "r"

The perimeter of square is given as:

Perimeter of square = 4a

The circumference of circle is given as:

Circumference of circle = 2πr

From given,

perimeter of a particular square and the circumference of a particular circle are equal

perimeter of a particular square = circumference of a particular circle

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

We need to find the ratio of the area of the square to the area of the circle.

We know that,

$\text{ Area of the square }= a^2\\\\\text{ Area of the circle }= \pi r^2$

Ratio of area of square to area of circle = $a^2 : \pi r^2$

Substituting the value of $a = \frac{\pi r}{2}$ in above ratio,

Ratio of area of square to area of circle = $(\frac{\pi r}{2})^2 : \pi r^2$

On reducing to lowest terms we get,

⇒ $\frac{\pi^2 r^2}{4} : \pi r^2 = \frac{\pi}{4} : 1$

In terms of fraction we get,

$\frac{\pi}{4} : 1 = \frac{\frac{\pi}{4}}{1} = \frac{\pi}{4}$

Thus ratio of the area of the square to the area of the circle is $\frac{\pi}{4} : 1$