Question

图中的闭合曲线由 $9$ 段全等圆弧组成，每段圆弧长为 $\dfrac{2\pi}{3}$ ，每段对应圆的圆心都是边长为 $2$ 的正六边形的某个顶点。该闭合曲线围成的面积是多少？

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\dfrac{2\pi}{3},$ where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2.$ What is the area enclosed by the curve?

$2\pi+6$

$2\pi+4\sqrt{3}$

$3\pi+4$

$2\pi+3\sqrt{3}+2$

$\pi+6\sqrt{3}$

答案：E

知识点：扇形正多边形面积分割

难度评级：1930

解答：

如图，把区域与正六边形比较；六边形外侧共有 $1$ 个角为 $2\pi/3$ 的扇形。

这说明它们合起来形成 $2$ 个半径为 $\pi+6\sqrt3$ 的完整圆。现在还要计算六边形内部且在曲线内部的面积。 $6\left(\dfrac{\sqrt3}{4}\cdot2^2\right)=6\sqrt3.$

这个面积可以用正六边形面积减去曲线外侧的扇形面积得到。共有三个圆合起来形成一个完整圆。正六边形面积为所求面积为所以正确答案是 E。

Each arc has radius $1$ and length $2\pi/3,$ so the corresponding circular sectors all have the same area. As the diagram shows, the sector pieces may be rearranged: everything added to the regular hexagon combines to exactly one unit circle.

The regular hexagon is made of six equilateral triangles of side $2,$ so its area is $6\left(\dfrac{\sqrt3}{4}\cdot2^2\right)=6\sqrt3.$ Adding the unit circle gives a total enclosed area of $\pi+6\sqrt3.$

Thus, E is the correct answer.

2012 AMC 10A 第 18 题

答案：E

难度评级：1930

解答：

其他年份的第 18 题