HMMT 二月 2005 · 冲刺赛 · 第 16 题

HMMT February 2005 — Guts Round — Problem 16

[8] A regular octahedron has a side length of 1. What is the distance between two opposite faces?

解析

A regular octahedron has a side length of 1. What is the distance between two opposite faces? √ Solution: 6 / 3 Imagine orienting the octahedron so that the two opposite faces are horizontal. Project onto a horizontal plane; these two faces are congruent equilateral triangles which (when projected) have the same center and opposite orientations. Hence, the vertices of the octahedron project to the vertices of a regular hexagon ABCDEF . F A O B E M C D Let O be the center of the hexagon and M the midpoint of AC . Now ABM is a √ √ √ 30-60-90 triangle, so AB = AM/ ( 3 / 2) = (1 / 2) / ( 3 / 2) = 3 / 3. If we let d denote the desired vertical distance between the opposite faces (which project to ACE and √ √ 2 2 2 2 BDF ), then by the Pythagorean Theorem, AB + d = 1 , so d = 1 − AB = 6 / 3.