AMC12 2002 A

AMC12 2002 A · Q5

AMC12 2002 A · Q5. It mainly tests Circle theorems, Area & perimeter.

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.

图中每个小圆的半径均为 1。最内侧的圆与围绕它的六个圆相切，而每个外侧小圆都与大圆及其相邻的小圆相切。求阴影区域的面积。

(A) $\pi$ $\pi$

(B) 1.5$\pi$ 1.5$\pi$

(D) 3$\pi$ 3$\pi$

(E) 3.5$\pi$ 3.5$\pi$

Answer

Correct choice: (C)

正确答案：（C）

Solution

The outer circle has radius $1+1+1=3$, and thus area $9\pi$. The little circles have area $\pi$ each; since there are 7, their total area is $7\pi$. Thus, our answer is $9\pi-7\pi=\boxed{2\pi\Rightarrow \textbf{(C)}}$.

外圆半径为 $1+1+1=3$，因此面积为 $9\pi$。每个小圆面积为 $\pi$；共有 7 个，所以总面积为 $7\pi$。因此答案为 $9\pi-7\pi=\boxed{2\pi\Rightarrow \textbf{(C)}}$。

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