AMC8 2010

AMC8 2010 · Q16

AMC8 2010 · Q16. It mainly tests Exponents & radicals, Area & perimeter.

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?

一个正方形和一个圆的面积相等。正方形边长与圆半径的比值为多少？

(A) $\sqrt{\frac{\pi}{2}}$ $\sqrt{\frac{\pi}{2}}$

(B) $\sqrt{\pi}$ $\sqrt{\pi}$

(D) $2\pi$ $2\pi$

(E) $\pi^2$ $\pi^2$

Answer

Correct choice: (B)

正确答案：（B）

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 $\frac{s^2}{r^2}=\pi$. Since $\left(\frac{s}{r}\right)^2=\frac{s^2}{r^2}$, we have $\frac{s}{r}=\sqrt{\pi}\Rightarrow \boxed{\textbf{(B)}\ \sqrt{\pi}}$

设正方形的边长为 $s$，圆的半径为 $r$。因此有 $s^2 = r^2 \pi$。两边除以 $r^2$，得到 $\frac{s^2}{r^2} = \pi$。由于 $\left( \frac{s}{r} \right)^2 = \frac{s^2}{r^2}$，因此 $\frac{s}{r} = \sqrt{\pi} \Rightarrow \boxed{\textbf{(B)}\ \sqrt{\pi}}$

Topics