2010 AMC 8 Problems/Problem 16

Problem

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? $\textbf{(A)}\ \frac{\sqrt{\pi}}{2}\qquad\textbf{(B)}\ \sqrt{\pi}\qquad\textbf{(C)}\ \pi\qquad\textbf{(D)}\ 2\pi\qquad\textbf{(E)}\ \pi^{2}$

Solution

Let the side length of the square be $s$ , and let the radius of the circle be $r$ . Thus we have $s^2=r^2\pi$ . Dividing each side by $r^2$ , we get $s^2/r^2=\pi$ . Since $(s/r)^2=s^2/r^2$ , we have $s/r=\sqrt{\pi}\Rightarrow \boxed{\textbf{(B)}\ \sqrt{\pi}}$

2010 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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2010 AMC 8 Problems/Problem 16

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