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 $\pi$.

Answer 1

Ratio of area of the square to the area of the circle = π/4

Step-by-step explanation:

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

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

Perimeter of square = 4 x a = 4a

Circumference of circle = 2πr

Given that

                      4a = 2πr

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

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

Area of the square = a²

Area of the circle = πr²

$\texttt{Ratio of area of the square to the area of the circle =}\frac{a^2}{\pi r^2}\\\\\texttt{Ratio of area of the square to the area of the circle =}\frac{\left ( \frac{\pi r}{2}\right )^2}{\pi r^2}\\\\\texttt{Ratio of area of the square to the area of the circle = }\frac{\pi}{4}$

Ratio of area of the square to the area of the circle = π/4