PUMaC 2009 · 几何（B 组） · 第 8 题

PUMaC 2009 — Geometry (Division B) — Problem 8

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the square of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator 1734 (e.g., if you think that the square is , then you would submit 1734274). 274 2

解析

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the square of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator 1734 (e.g., if you think that the square is , then you would submit 1734274). 274 Solution. 52972. Extend the edges that are common to two hexagons. We obtain a regular tetrahedron of side length 3. Hence the volume of original solid is a regular tetrahedron of side length 3 minus volume of 4 regular tetrahedrons of side length 1. The volume is √ √ √ 1 9 3 27 − 4 23 2 × × 6 × = 3 4 27 12 . 3