What is the angle between any two legs of a regular tetrahedron? (answer: ~109.47… deg) The question arises in many situations. For example, what is the angle between carbon atoms in diamond? Here we take advantage of the symmetry of the tetrahedron to find this angle.

We can find h in a roundabout way by, taking advantage of the symmetry of the tetrahedron. Break the big tetrahedron into 4 smaller tetrahedra.

## Thoughts for the reader:

What about this discussion changes for higher dimensional analogues of tetrahedra (ie: regular n-simplices)?

As the dimension of the simplex increases, what does the angle go towards (answer: 90 degrees)?

The maximum number of mutually orthogonal nonzero vectors in n-dimensional space is n vectors. Suppose we relax this a little bit. Is it possible to find n+1 “approximately orthogonal” vectors in n-dimensional space?

## See also:

C. Giomini, G. Marrosu, M.E. Cardinali:

The exploded tetrahedron.

(Educ. Chem., 1995*, 32*, p. 38)

Exploded_tetrahedron.pdf (pdf of Giomini et al, provided by author)

http://chimianet.zefat.ac.il/download/Tetrahedral_angle.doc (Hebraic characters)

http://mathcentral.uregina.ca/QQ/database/QQ.09.00/nishi1.html (alternate geometric derivation involving a cube)

dont get the equation

Which equation?

Exactly what I was looking for. Awesome!

Great! This really helped with my math project!

hi??

Hi?

why does cos of theta=-h?

The cosine of the supplementary angle, cos(theta_s), is equal to h by the diagram and rules of cosines (adjacent/hypotenuse = h/1 = h). When you change from the supplementary angle to the original angle, you have to change the sign, thus cos(theta) = -h.