If the perimeter of a circle is equal to the perimeter of a square, find the ratio of the area of the square to the area of the circle.

Step-by-step solution

1. Step 1: Write the perimeter of a circle: $P_{\text{circle}}=2\pi r$.
2. Step 2: Write the perimeter of a square: $P_{\text{square}}=4a$.
3. Step 3: Set them equal: $2\pi r=4a\to a=\frac{\pi r}{2}$.
4. Step 4: Compute area of square: $A_{\text{square}}=a^2={\left(\frac{\pi r}{2}\right)}^2=\frac{{\pi }^2{r}^2}{4}$.
5. Step 5: Compute area of circle: $A_{\text{circle}}=\pi r^2$.
6. Step 6: Ratio $\frac{A_{\text{square}}}{A_{\text{circle}}}=\frac{\frac{{\pi }^2{r}^2}{4}}{\pi r^2}=\frac{\pi }{4}$.
7. Step 7: The required ratio is $\frac{\pi }{4}$.

Concept tested: Perimeter equality to compare areas

Published 16 October 2026 · How this solution was prepared and checked