AMC 8 2012 Problem 24

AMC8 2012 年第 24 题

圆★★★★☆

A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?

半径为 2 的圆被分割成 4 段相等的弧，这 4 段弧拼接成如图所示的星型。那么这个星型的面积和原来圆的面积的比值是多少？

A. $\displaystyle \hspace{.05in}\frac{4-\pi}{\pi}$
B. $\displaystyle \hspace{.05in}\frac{1}\pi$
C. $\displaystyle \hspace{.05in}\frac{\sqrt2}{\pi}$
D. $\displaystyle \hspace{.05in}\frac{\pi-1}{\pi}$
E. $\displaystyle \hspace{.05in}\frac{3}\pi$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 外接正方形在星形外画一个正方形，边长等于圆的直径 4。四个角恰好是四个四分之一圆，合起来是一个半径 2 的整圆，面积

\pi\cdot2^2=4\pi

。正方形面积 16，故星形面积

16-4\pi

。原圆面积

4\pi

，比值

\dfrac{16-4\pi}{4\pi}=\dfrac{4-\pi}{\pi}

。

Solution 1

Elementary · circumscribing square Draw a square around the star with side 4 (the circle's diameter). Its four corners are quarter-circles that together make one full circle of radius 2, area

4\pi

. So the star's area is

16-4\pi

, and the ratio to the original circle (

4\pi

) is

\dfrac{16-4\pi}{4\pi}=\dfrac{4-\pi}{\pi}

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