AMC 8 2011 Problem 25

AMC8 2011 年第 25 题

圆★★★★☆

A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?

一个半径为 1 的圆内切于一个正方形中，并且外接于另一个正方形，如图所示。那么圆的阴影部分的面积和两个正方形之间的面积的比值最接近下面哪个分数？

A. $\displaystyle \frac{1}2$
B.1
C. $\displaystyle \frac{3}2$
D.2
E. $\displaystyle \frac{5}2$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 算面积外正方形边长 2、面积 4；内正方形对角线等于直径 2、面积

\displaystyle \tfrac{2^2}{2}=2

，故两方之间面积 4-2=2。圆的阴影部分

=\pi-2

。比值

\dfrac{\pi-2}{2}\approx\dfrac{1.14}{2}\approx0.57

，最接近

\dfrac12

。

Solution 1

Elementary · compute the area The outer square has side 2 and area 4; the inner square has diagonal equal to the diameter 2, so area

\displaystyle \tfrac{2^2}{2}=2

, and the area between the squares is 4-2=2. The circle's shaded area is

\pi-2

. The ratio is

\dfrac{\pi-2}{2}\approx\dfrac{1.14}{2}\approx0.57

, closest to

\tfrac12

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