Question

一个等边三角形和一个正六边形周长相等。如果三角形面积为 $4$ ，那么六边形面积是多少？

An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is $4$ , what is the area of the hexagon?

$4$

$5$

$6$

$4\sqrt3$

$6\sqrt3$

答案：C

知识点：等边三角形正多边形面积比

难度评级：1540

视频讲解：

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文字解答：

设三角形边长为 $s$ 。其周长为 $3s$ ，所以六边形边长为 $\frac{3s}{6} = \frac s2$ 。

如图，正六边形可分成 $6$ 个边长为 $\dfrac{s}{2}$ 的等边三角形。每个小三角形是原三角形按比例 $\dfrac{1}{2}$ 缩小得到的，所以面积缩小为 $(\dfrac{1}{2})^2 = \dfrac{1}{4}$ 。因此每个小三角形面积为 $4 \cdot \dfrac{1}{4} = 1$ 。共有 $6$ 个，所以六边形面积为 $6\cdot 1 = 6$ 。

所以正确答案是 C。

Let the side length of the triangle be $s.$ This means the perimeter is $3s.$ Therefore, the side length for the hexagon is $\frac{3s}{6} = \frac s2.$

A hexagon can be made of $6$ equilateral triangles with side length $\dfrac{s}{2}$ as shown above. Each triangle is the original triangle scaled down by $\dfrac{1}{2},$ so the area is scaled down by $(\dfrac{1}{2})^2 = \dfrac{1}{4} .$ Therefore, the area of each of these triangles is $4 \cdot \dfrac{1}{4} = 1.$ Since there are $6$ of them, the area is $6\cdot 1 = 6.$

Thus, the answer is C .

2012 AMC 8 第 23 题

答案：C

难度评级：1540

视频讲解：

文字解答：

其他年份的第 23 题