2012 AMC 8 第 23 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
23.
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 ，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 , what is the area of the hexagon?
小提示：
周长相等使六边形边长为三角形边长的一半。
Equal perimeters make the hexagon side half the triangle side.
大提示：
正六边形由六个全等等边三角形组成。
A regular hexagon is made from six congruent equilateral triangles.
视频讲解：
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文字解答：
设三角形边长为 。其周长为 ，所以六边形边长为 。
如图，正六边形可分成 个边长为 的等边三角形。每个小三角形是原三角形按比例 缩小得到的，所以面积缩小为 。因此每个小三角形面积为 。共有 个，所以六边形面积为 。
所以正确答案是 C。
Let the side length of the triangle be This means the perimeter is Therefore, the side length for the hexagon is
A hexagon can be made of equilateral triangles with side length as shown above. Each triangle is the original triangle scaled down by so the area is scaled down by Therefore, the area of each of these triangles is Since there are of them, the area is
Thus, the answer is C .
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