AMC8 2012

AMC8 2012 · Q23

AMC8 2012 · Q23. It mainly tests Triangles (properties), Similarity.

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

一个等边三角形和一个正六边形具有相同的周长。如果三角形的面积是4，则六边形的面积是多少？

(A) 4 4

(B) 5 5

(D) $4\sqrt{3}$ $4\sqrt{3}$

(E) $6\sqrt{3}$ $6\sqrt{3}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let the perimeter of the equilateral triangle be $3s$. The side length of the equilateral triangle would then be $s$ and the sidelength of the hexagon would be $\frac{s}{2}$. A hexagon contains six equilateral triangles. One of these triangles would be similar to the large equilateral triangle in the ratio $1 : 4$, since the sidelength of the small equilateral triangle is half the side length of the large one. Thus, the area of one of the small equilateral triangles is $1$. The area of the hexagon is then $1 \times 6 = \boxed{\textbf{(C)}\ 6}$.

令等边三角形的周长为$3s$。则等边三角形的边长为$s$，六边形的边长为$\frac{s}{2}$。六边形包含六个等边三角形。其中一个三角形与大等边三角形相似，比率为$1 : 2$（因为小三角形的边长是大三角形边长的一半，面积比为$1:4$）。大三角形面积为4，故小等边三角形面积为$1$。因此六边形面积为$1 \times 6 = \boxed{\textbf{(C)}\ 6}$。

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