Question

大圆的直径是 $AC$ 。两个小圆的圆心在 $AC$ 上，并且在大圆圆心 $O$ 处相切。如果每个小圆半径为 $1$ ，阴影区域面积与一个小圆面积之比是多少？

The large circle has diameter $AC.$ The two small circles have their centers on $AC$ and just touch at $O,$ the center of the large circle. If each small circle has radius $1,$ what is the value of the ratio of the area of the shaded region to the area of one of the small circles?

在 $\dfrac12$ 和 $1$ 之间

between $\dfrac12$ and $1$

$1$

在 $1$ 和 $\dfrac32$ 之间

between $1$ and $\dfrac32$

在 $\dfrac32$ 和 $2$ 之间

between $\dfrac32$ and $2$

无法由给定信息确定

cannot be determined from the information given

答案：B

知识点：圆面积面积分割

难度评级：1090

解答：

大圆半径为 $2$ ，所以面积是 $\pi(2)^2 = 4\pi$ ，每个小圆面积是 $\pi$ 。

由对称性，阴影区域是面积差的一半： $\dfrac12(4\pi - 2\pi) = \pi$ 。它与一个小圆面积 $\pi$ 的比值是 $1$ 。

所以正确答案是 B。

The large circle has radius $2,$ so its area is $\pi(2)^2 = 4\pi,$ and each small circle has area $\pi.$

By symmetry the shaded region is half the difference of the areas: $\dfrac12(4\pi - 2\pi) = \pi.$ The ratio of this to one small circle's area $\pi$ is $1.$

Thus, the correct answer is B .

1986 AMC 8 第 23 题

答案：B

难度评级：1090

解答：

其他年份的第 23 题