HMMT 二月 2004 · 冲刺赛 · 第 27 题

英文解析

27. A regular hexagon has one side along the diameter of a semicircle, and the two oppositevertices on the semicircle. Find the area of the hexagon if the diameter of the semicircleis 1.
    √
    Solution: 3 3 / 26
    The midpoint of the side of the hexagon on the diameter is the center of the circle.
    Draw the segment from this center to a vertex of the hexagon on the circle. Thissegment, whose length is 1 / 2, is the hypotenuse of a right triangle whose legs have
    √
    lengths a/ 2 and a 3, where a is a side of the hexagon. So 1 / 4 = a (1 / 4 + 3), so 2
    a = 1 / 13. The hexagon consists of 6 equilateral triangles of side length a , so the area 2
    √ √
    of the hexagon is 3 a 3 / 2 = 3 3 / 26.2
    7 a