# Tetrahedron angles

**URL:** <https://boards.straightdope.com/t/tetrahedron-angles/317613>\
**Category:** Factual Questions\
**Created:** [August 18, 2005, 5:46pm UTC](https://boards.straightdope.com/t/tetrahedron-angles/317613 "2005-08-18T17:46:29Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![quelquechose](https://avatars.discourse-cdn.com/v4/letter/q/6f9a4e/32.png) [@quelquechose](https://boards.straightdope.com/u/quelquechose)\
**Post date:** [August 18, 2005, 5:46pm UTC](https://boards.straightdope.com/t/tetrahedron-angles/317613/1 "2005-08-18T17:46:29Z")

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In chemistry the angle 109.5 comes up a lot, as the angle from one corner of a regular tetrahedron to the center to another corner, as in a methane molecule. So, while I was waiting for the bus yesterday I tried to derive that angle by myself, and it turned out to be far more difficult than I expected. I used the projection of the tetrahedron on a plane parallel to one face and a cross section through two of the center-to-corner segments to get three equations and three variables, which I then solved for the angle I wanted. I got the right answer, but I’d like to know if there’s an easier way to do this. I googled a few ways of solving the problem and found slightly simpler methods to solve this problem, but nothing that was much simpler. One idea I had was to find two vectors parallel to the chemical bonds in CH[sub]4[/sub] and determine the angle between them, but thus far I’ve been unsuccessful. So, is there any easy way to find an exact solution to this problem?

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**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\
**Post date:** [August 18, 2005, 6:17pm UTC](https://boards.straightdope.com/t/tetrahedron-angles/317613/2 "2005-08-18T18:17:17Z")

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> [@quelquechose](#):
>
> In chemistry the angle 109.5 comes up a lot, as the angle from one corner of a regular tetrahedron to the center to another corner, as in a methane molecule. … So, is there any easy way to find an exact solution to this problem?

The tetrahedron with vertices (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1) is a regular tetrahedron centered at the origin. So you can just figure out the angle between vectors [1,1,1] and [1,-1,-1].

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**Author:** ![CJJ](https://avatars.discourse-cdn.com/v4/letter/c/ecc23a/32.png) [@CJJ](https://boards.straightdope.com/u/CJJ)\
**Post date:** [August 18, 2005, 7:01pm UTC](https://boards.straightdope.com/t/tetrahedron-angles/317613/3 "2005-08-18T19:01:43Z")

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> [@](#):
>
> The tetrahedron with vertices (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1) is a regular tetrahedron centered at the origin. So you can just figure out the angle between vectors [1,1,1] and [1,-1,-1].

To complete the thought, the dot product between these two cartesian vectors is 1_1+1_-1+1\*-1 = -1. This is equal to the product of the vector’s magnitudes (each magnitude = sqrt(3)) times the cosine of the angle between them. Thus cos(th) = -(1/3), so th = 109.47 degrees.

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**Author:** ![quelquechose](https://avatars.discourse-cdn.com/v4/letter/q/6f9a4e/32.png) [@quelquechose](https://boards.straightdope.com/u/quelquechose)\
**Post date:** [August 18, 2005, 7:38pm UTC](https://boards.straightdope.com/t/tetrahedron-angles/317613/4 "2005-08-18T19:38:30Z")

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Thanks, that’s exactly what I was looking for.