AMC8 2012

AMC8 2012 · Q24

AMC8 2012 · Q24. It mainly tests Circle theorems, Area & perimeter.

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?

一个半径为2的圆被切成四个全等的弧。这四个弧连接形成图示的星形图形。星形图形的面积与原圆面积的比率为多少？

(A) $\frac{4-\pi}{\pi}$ $\frac{4-\pi}{\pi}$

(B) $\frac{1}{\pi}$ $\frac{1}{\pi}$

(D) $\frac{\pi-1}{\pi}$ $\frac{\pi-1}{\pi}$

(E) $\frac{3}{\pi}$ $\frac{3}{\pi}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Draw a square around the star figure. The side length of this square is $4$, because the side length is the diameter of the circle. The square forms $4$-quarter circles around the star figure. This is the equivalent of one large circle with radius $2$, meaning that the total area of the quarter circles is $4\pi$. The area of the square is $16$. Thus, the area of the star figure is $16 - 4\pi$. The area of the circle is $4\pi$. Taking the ratio of the two areas, we find the answer is $\boxed{\textbf{(A)}\ \frac{4-\pi}{\pi}}$.

在星形图形外画一个正方形。这个正方形的边长为$4$，因为边长等于圆的直径。正方形围绕星形图形形成了$4$个四分之一圆。这相当于一个半径为$2$的大圆，总面积为$4\pi$。正方形面积为$16$。因此，星形图形的面积为$16 - 4\pi$。原圆面积为$4\pi$。两面积之比为$\boxed{\textbf{(A)}\ \frac{4-\pi}{\pi}}$。

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