1952 AMC 12 Problem 33

Question

A circle and a square have the same perimeter. Then:

Their areas are equal

The area of the circle is the greater

The area of the square is the greater

The area of the circle is

\pi

times the area of the square

None of these

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.

Problem 33 in other years