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

小提示：

每段圆弧都是单位圆上的 $120^\circ$ 弧。

Each arc is a $120^\circ$ arc of a unit circle

大提示：

把这些扇形块围绕正六边形重新拼接。

Rearrange the sector pieces around the regular hexagon

解答：

每段圆弧的半径都是 $1$ ，长度都是 $\frac{2\pi}{3}$ ，所以对应的各个扇形面积都相同。如图所示，这些扇形块可以重新拼接：添加到正六边形上的部分合起来恰好构成一个单位圆。

正六边形由六个边长为 $2$ 的等边三角形拼成，所以它的面积为 $6\left(\dfrac{\sqrt3}{4}\cdot2^2\right)=6\sqrt3\text{。}$ 再加上那个单位圆，围成的总面积为 $\pi+6\sqrt3$ 。

所以正确答案是 E。

Each arc has radius $1$ and length $\frac{2\pi}{3},$ 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

小提示：

大提示：

解答：

其他年份的第 18 题