HMMT 二月 2004 · 冲刺赛 · 第 33 题

英文解析

33. A plane P slices through a cube of volume 1 with a cross-section in the shape of aregular hexagon. This cube also has an inscribed sphere, whose intersection with Pis a circle. What is the area of the region inside the regular hexagon but outside the circle?
    √
    Solution: (3 3 − π ) / 4
    One can show that the hexagon must have as its vertices the midpoints of six edges ofthe cube, as illustrated; for example, this readily follows from the fact that oppositesides of the hexagons and the medians between them are parallel. We then conclude
    √
    that the side of the hexagon is 2 / 2 (since it cuts off an isosceles triangle of leg 1 / 2
    √ √ √
    from each face), so the area is (3 / 2)( 2 / 2) ( 3) = 3 3 / 4. Also, the plane passes 2
    through the center of the sphere by symmetry, so it cuts out a cross section of radius
    1 / 2, whose area (which is contained entirely inside the hexagon) is then π/ 4. The
    √
    sought area is thus (3 3 − π ) / 4.