1970 AMC 12 Problem 2

Attempt Problem 2 of the 1970 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1970 AMC 12 solutions, or check the answer key.

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A square and a circle have equal perimeters. The ratio of the area of the circle to the area of the square is:

\frac{4}{\pi}

\frac{\pi}{\sqrt2}

\frac{4}{1}

\frac{\sqrt2}{\pi}

\frac{\pi}{4}

Let the common perimeter be

p

Express the circle radius and square side in terms of

p

, then divide their areas

If the common perimeter is

p,

then the circle has radius

\frac{p}{2\pi}

and the square has side

\frac{p}{4}.

Therefore

\frac{\pi\left(\frac{p}{2\pi}\right)^2}{\left(\frac p4\right)^2} =\frac{\frac{p^2}{4\pi}}{\frac{p^2}{16}}=\frac4\pi.

Therefore, the correct answer is A.

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