Question

一个半径为二的圆被分成四段全等圆弧。将这四段圆弧连接成图中的星形。星形面积与原圆面积之比是多少？

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?

$\dfrac{4-\pi}{\pi}$

$\dfrac{1}\pi$

$\dfrac{\sqrt2}{\pi}$

$\dfrac{\pi-1}{\pi}$

$\dfrac{3}\pi$

答案：A

知识点：圆面积面积分割

难度评级：1860

视频讲解：

Click to load, then click again to play

文字解答：

原圆面积为 $\pi\cdot2^2=4\pi$ 。

连接四个四分之一圆弧端点形成一个正方形。这个正方形的两条对角线都是 $4$ 和 $4$ ，所以面积为 $\frac12\cdot4\cdot4=8$ 。

圆内但正方形外的部分面积为 $4\pi-8$ 。这四块与正方形内但星形外的四块全等。

因此星形面积为 $8-(4\pi-8)=16-4\pi$ 。所求比值为 $\dfrac{16-4\pi}{4\pi}=\dfrac{4-\pi}{\pi}$ 。所以正确答案是 A。

The area of the original circle is $\pi\cdot2^2=4\pi$ .

Join the four quarter-circle endpoints to form a square. The square has diagonals $4$ and $4$ , so its area is $\frac12\cdot4\cdot4=8$ .

The part inside the circle but outside this square has area $4\pi-8$ . Those four pieces are congruent to the pieces inside the square but outside the star.

Thus the star area is $8-(4\pi-8)=16-4\pi$ . The desired ratio is $\dfrac{16-4\pi}{4\pi}=\dfrac{4-\pi}{\pi}$ . Thus, the answer is A .

2012 AMC 8 第 24 题

答案：A

难度评级：1860

视频讲解：

文字解答：

其他年份的第 24 题