3321.11 – A Circle and A Square

Suppose a circle and a square have the same area. What is the ratio of the area of a square inscribed inside the circle to the area of a circle inscribed inside the square?

Solution

Referring to the diagram below, we are given that $\pi r^s = s^2$ so that $r^2 = s^2/\pi$ . We seek the ratio $A_1/A_2$ . We have,

\begin{aligned} A_1 &=& \text{ half the product of the diagonals } \\ &=& \frac{1}{2} (2r \cdot 2r) = 2r^2, \\ A_2 &=& \pi \left( \frac{s}{2} \right)^2 = \frac{\pi s^2}{4}, \\ \frac{A_1}{A_2} &=& 2r^2 / \frac{\pi s^2}{4} = \frac{2 s^2}{\pi} / \frac{\pi s^2}{4}\\ &=& \frac{2 s^2}{\pi} \cdot \frac{4}{\pi s^2} = \frac{8}{\pi^2}. \end{aligned}