AMC10 2014 A

AMC10 2014 A · Q12

AMC10 2014 A · Q12. It mainly tests Triangles (properties), Area & perimeter.

A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region?

一个边长为6的正六边形。以每个顶点为中心，画出半径为3的圆弧，形成如图所示的扇形。六边形内部但扇形外部的区域被涂阴影如图所示。阴影区域的面积是多少？

(A) $27\sqrt{3} - 9\pi$ $27\sqrt{3} - 9\pi$

(B) $27\sqrt{3} - 6\pi$ $27\sqrt{3} - 6\pi$

(D) $54\sqrt{3} - 12\pi$ $54\sqrt{3} - 12\pi$

(E) $108\sqrt{3} - 9\pi$ $108\sqrt{3} - 9\pi$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Each of the 6 sectors has radius 3 and central angle $120^\circ$. Their combined area is $6\cdot \frac{1}{3}\cdot \pi \cdot 3^2 = 18\pi$. The hexagon can be partitioned into 6 equilateral triangles each having side length 6, so the hexagon has area $6\cdot \frac{\sqrt{3}}{4}\cdot 6^2 = 54\sqrt{3}$. The shaded region has area $54\sqrt{3}-18\pi$.

答案（C）：6 个扇形的半径都是 3，圆心角为 $120^\circ$。它们的总面积为 $6\cdot \frac{1}{3}\cdot \pi \cdot 3^2 = 18\pi$。该六边形可以分成 6 个边长为 6 的正三角形，因此六边形的面积为 $6\cdot \frac{\sqrt{3}}{4}\cdot 6^2 = 54\sqrt{3}$。阴影部分的面积为 $54\sqrt{3}-18\pi$。

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