AMC10 2002 A

AMC10 2002 A · Q5

AMC10 2002 A · Q5. It mainly tests Triangles (properties), 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.

图中小圆的半径均为 1。最内侧的圆与围绕它的六个圆相切，每个那些圆与大圆及其小圆邻居相切。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

(C) The large circle has radius $3$, so its area is $\pi\cdot 3^2 = 9\pi$. The seven small circles have a total area of $7(\pi\cdot 1^2)=7\pi$. So the shaded region has area $9\pi-7\pi=2\pi$.

（C）大圆的半径为 $3$，所以它的面积是 $\pi\cdot 3^2 = 9\pi$。七个小圆的总面积是 $7(\pi\cdot 1^2)=7\pi$。因此，阴影部分的面积为 $9\pi-7\pi=2\pi$。

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