HMMT 二月 2005 · 冲刺赛 · 第 30 题

英文解析

30. A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles suchthat two squares and two triangles alternate around each vertex, as shown.
    What is the volume of a cuboctahedron of side length 1?
    √
    Solution: 5 2 / 3
    We can construct a cube such that the vertices of the cuboctahedron are the midpointsof the edges of the cube.
    Let s be the side length of this cube. Now, the cuboctahedron is obtained from thecube by cutting a tetrahedron from each corner. Each such tetrahedron has a base inthe form of an isosceles right triangle of area ( s/ 2) / 2 and height s/ 2 for a volume of 2
    ( s/ 2) / 6. The total volume of the cuboctahedron is therefore 3
    3 3 3
    s − 8 · ( s/ 2) / 6 = 5 s / 6 .
    Now, the side of the cuboctahedron is the hypotenuse of an isosceles right triangle
    √ √
    of leg s/ 2; thus 1 = ( s/ 2) 2, giving s = 2, so the volume of the cuboctahedron is
    √
    5 2 / 3.
    11