Angles of an Octahedron

Date: 07/20/98 at 18:33:51
From: Phani
Subject: Octahedron

I wanted to know the angle between two adjacent faces of an 
octahedron.

Date: 07/22/98 at 11:27:54
From: Doctor Rick
Subject: Re: Octahedron

Hello, Phani. I assume you refer to a regular octahedron. It helps me 
to have a model of the regular octahedron in front of me. I have one 
that I made of index cards. Another can be seen at:

http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html   

One way to compute the angle you want is to notice that the centers of 
the faces of a regular octahedron are the vertices of a cube, and that 
the diagonals of the cube are perpendicular to the faces of the 
octahedron. The angle between adjacent faces is the supplement of the 
angle between normals to the faces, which is the angle between 
diagonals of a cube. This angle can be computed most easily using 
vectors: make a cube with corners (+/-1, +/-1, +/-1). That's plus or 
minus 1 for each coordinate, giving 8 points. You want to find the 
angle between vectors A = (1, 1, 1) and B = (1, 1, -1), and you can
do this using the rule that the dot product A.B = |A|*|B|*cos(angle). 
Then subtract this angle from 180 degrees.

If you make a model of a tetrahedron with the same edge length as the
octahedron, you will notice that the angle you are after is the 
supplement of the angle between faces of the tetrahedron.This angle in 
turn is the supplement of the "tetrahedral angle" that organic chemists 
care about - the angle at the center between two vertices of the 
tetrahedron. You can see this by the same argument I made above about 
faces and normals. And the upshot of this observation is that the angle 
you want is just the tetrahedral angle. If you don't want to do the 
calculation, just ask a chemist.

Write back if you need more explanation.

- Doctor Rick, The Math Forum
Check out our web site! http://mathforum.org/dr.math/