AMC 8 2005 Problem 25

AMC8 2005 年第 25 题

圆★★★★☆

A square with side length 2.0 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，两者中心重合。位于圆内且在正方形之外的区域的面积，等于位于圆外且在正方形之内的区域的面积。那么圆的半径是多少？

A. $\displaystyle \frac{2}{\sqrt{\pi}}$
B. $\displaystyle \frac{1+\sqrt{2}}{2}$
C. $\displaystyle \frac{3}{2}$
D. $\sqrt{3}$
E. $\sqrt{\pi}$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 设公共面积设圆与正方形重叠部分的面积为

a

。在圆内、方外的面积是

\pi r^2-a

；在圆外、方内的面积是

2^2-a=4-a

。两者相等

\pi r^2-a=4-a

，约去

a

得

\pi r^2=4,\qquad r=\dfrac{2}{\sqrt\pi}

Solution 1

Elementary · set the common area Let

a

be the area inside both the circle and the square. The region inside the circle but outside the square has area

\pi r^2-a

, and the region outside the circle but inside the square has area

2^2-a=4-a

. Setting them equal,

\pi r^2-a=4-a

, so

\pi r^2=4,\qquad r=\dfrac{2}{\sqrt\pi}.

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