2002 AMC 12A 第 5 题

Question

图中每个小圆的半径都是一。最里面的圆与围绕它的六个圆相切，这六个圆中的每个又与大圆以及相邻的小圆相切。求阴影区域的面积。

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

\pi

1.5\pi

2\pi

3\pi

3.5\pi

答案：C

知识点：圆面积相切圆

难度评级：1270

小提示：

每个外圈圆的圆心到中心的距离为

2

，所以大圆半径为

3

。

Each surrounding circle has its center

2

units from the center, so the large circle has radius

3

大提示：

用大圆面积减去七个单位圆的面积。

Subtract the areas of the seven unit circles from the area of the large circle

解答：

六个外圈单位圆都与中心单位圆相切，所以它们的圆心到中心距离为

2

。再加上一个半径，大圆半径为

3

，面积为

9\pi

。

七个单位圆的总面积为

7\pi

，所以阴影区域面积为

9\pi - 7\pi = 2\pi

。

因此，正确答案是 C。

Each of the six outer unit circles is tangent to the central unit circle, so its center is

2

units from the center. Adding one more radius, the large circle has radius

3

and area

9\pi.

The seven unit circles have total area

7\pi,

so the shaded region has area

9\pi - 7\pi = 2\pi.

Thus, the correct answer is C.

2002 AMC 12A 第 5 题

答案：C

难度评级：1270

小提示：

大提示：

解答：

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