1952 AMC 12 第 33 题

Call the common perimeter $P$ and express each area in terms of $P$

The circle has area $\frac{P^2}{4\pi}$, while the square has area $\frac{P^2}{16}$

With common perimeter $P,$ the circle’s radius is $\frac{P}{2\pi}$, so its area is $\frac{P^2}{4\pi}.$ The square’s side is $\frac{P}{4},$ so its area is $\frac{P^2}{16}.$ Since $4\pi\lt16,$ $$\frac{P^2}{4\pi}\gt\frac{P^2}{16}.$$

The circle has the greater area, so the correct answer is B.