HMMT 二月 2005 · 几何 · 第 2 题

HMMT February 2005 — Geometry — Problem 2

Let ABCD be a regular tetrahedron with side length 2. The plane parallel to edges AB and CD and lying halfway between them cuts ABCD into two pieces. Find the surface area of one of these pieces.

解析

Let ABCD be a regular tetrahedron with side length 2. The plane parallel to edges AB and CD and lying halfway between them cuts ABCD into two pieces. Find the surface area of one of these pieces. √ Solution: 1 + 2 3 The plane intersects each face of the tetrahedron in a midline of the face; by symmetry it follows that the intersection of the plane with the tetrahedron is a square of side length 1. The surface area of each piece is half the total surface area of the tetrahedron √ √ 2 1 2 3 plus the area of the square, that is, · 4 · + 1 = 1 + 2 3. 2 4