AMC10 2008 A

AMC10 2008 A · Q17

AMC10 2008 A · Q17. It mainly tests Area & perimeter, Geometric probability (basic).

An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the triangle and not more than 3 units from a point of the triangle?

一个边长为 6 的等边三角形。求三角形外部且距离三角形某点不超过 3 个单位的区域的面积。

(A) $36 + 24\sqrt{3}$ $36 + 24\sqrt{3}$

(B) $54 + 9\pi$ $54 + 9\pi$

(D) $(2\sqrt{3} + 3)^2 \pi$ $(2\sqrt{3} + 3)^2 \pi$

(E) $9(\sqrt{3} + 1)^2 \pi$ $9(\sqrt{3} + 1)^2 \pi$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): The region consists of three rectangles with length 6 and width 3 together with three 120° sectors of circles with radius 3. The combined area of the three 120° sectors is the same as the area of a circle with radius 3, so the area of the region is $3\cdot 6\cdot 3+\pi\cdot 3^2=54+9\pi.$

答案（B）：该区域由三个长为 6、宽为 3 的长方形，以及三个半径为 3 的圆的 $120^\circ$ 扇形组成。三个 $120^\circ$ 扇形的总面积等于一个半径为 3 的圆的面积，因此该区域的面积为 $3\cdot 6\cdot 3+\pi\cdot 3^2=54+9\pi.$

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