1970 AMC 12 第 24 题

Question

一个等边三角形与一个正六边形的周长相等。若三角形的面积为

2

，则六边形的面积为：

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

2,

then the area of the hexagon is:

2

3

4

6

12

设六边形的边长为

s

；由周长相等可知三角形的边长为

2s

Let the hexagon side be

s

; equal perimeters make the triangle side

2s

三角形可分成四个边长为

s

的小等边三角形，而六边形可分成六个

The triangle splits into four small equilateral triangles of side

s

, while the hexagon splits into six

设六边形的边长为

s

。它的周长为

6s

，所以等边三角形的边长为

2s

。大三角形由

4

个边长为

s

的等边三角形组成，而六边形由

6

个这样的三角形组成。因此它们的面积之比为

\frac{6}{4}=\frac{3}{2}

。由于三角形的面积为

2

，所以六边形的面积为

3

。

因此，正确答案是 B。

Let the hexagon side be

s.

Its perimeter is

6s,

so the equilateral triangle has side

2s.

The large triangle consists of

4

equilateral triangles of side

s,

while the hexagon consists of

6.

Their area ratio is therefore

\frac{6}{4}=\frac{3}{2}.

Since the triangle’s area is

2,

the hexagon’s area is

3.

Therefore, the correct answer is B.

其他年份的第 24 题