Question

一个正六边形边长为 $6$ 。以每个顶点为圆心画半径为 $3$ 的全等圆弧，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。阴影区域的面积是多少？

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?

$27\sqrt{3}-9\pi$

$27\sqrt{3}-6\pi$

$54\sqrt{3}-18\pi$

$54\sqrt{3}-12\pi$

$108\sqrt{3}-9\pi$

答案：C

知识点：正多边形扇形面积

难度评级：1370

解答：

$6$ $6$ 。

$s$ $\dfrac{s^2 \sqrt{3}}{4}.$

正六边形可分成个边长为的等边三角形。 $6 \cdot \dfrac{6^2 \sqrt{3}}{4} = 54\sqrt{3}.$

边长为 $120^{\circ}$ 的等边三角形面积公式为 $2$

正六边形每个内角为，六个扇形合起来相当于个完整圆。 $2 \cdot 3^2 \pi = 18\pi.$

所以阴影区域的面积为 $54\sqrt{3} - 18\pi.$

所以正确答案是 C。

Note that we can split the hexagon up into $6$ equilateral triangles each with side length $6.$

Recall that the area of an equilateral triangle with side length $s$ is $\dfrac{s^2 \sqrt{3}}{4}.$

This means that the area of the hexagon is $6 \cdot \dfrac{6^2 \sqrt{3}}{4} = 54\sqrt{3}.$

Since each interior angle of a regular hexagon is $120^{\circ},$ the six sectors form $2$ full circles.

This means that the area of all the sectors is $2 \cdot 3^2 \pi = 18\pi.$

The area of the shaded region is then $54\sqrt{3} - 18\pi.$

Thus, C is the correct answer.

2014 AMC 10A 第 12 题

答案：C

难度评级：1370

解答：

其他年份的第 12 题