AMC8 2005

AMC8 2005 · Q25

AMC8 2005 · Q25. It mainly tests Linear equations, Area & perimeter.

A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

一个边长为2的正方形和一个圆同心。圆内、正方形外的区域总面积等于圆外、正方形内的区域总面积。求圆的半径。

(A) $\frac{2}{\sqrt{\pi}}$ 2/√π

(B) $\frac{1+\sqrt{2}}{2}$ (1+√2)/2

(D) $\sqrt{3}$ √3

(E) $\sqrt{\pi}$ √π

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) Because the circle and square share the same interior region and the area of the two exterior regions indicated are equal, the square and the circle must have equal area. The area of the square is $2^2$ or $4$. Because the area of both the circle and the square is $4$, $4=\pi r^2$. Solving for $r$, the radius of the circle, yields $r^2=\frac{4}{\pi}$, so $r=\sqrt{\frac{4}{\pi}}=\frac{2}{\sqrt{\pi}}$. Note: It is not necessary that the circle and square have the same center.

（A）因为圆和正方形共享同一个内部区域，并且图中标出的两个外部区域面积相等，所以正方形和圆的面积必须相等。正方形的面积是$2^2$，即$4$。由于圆和正方形的面积都为$4$，有$4=\pi r^2$。解出圆的半径$r$，得到$r^2=\frac{4}{\pi}$，因此$r=\sqrt{\frac{4}{\pi}}=\frac{2}{\sqrt{\pi}}$。注：圆和正方形不一定需要有相同的圆心（中心）。

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