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### 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

Write back if you need more explanation.

- Doctor Rick, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Polyhedra

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