During dinner tonight, Hon Leung, Ken and I were discussing some possible exercises for the bright secondary school students. I suggested generalizing the cosine law

for a triangle to a tetrahedron or even an -dimensional simplex. I think this would be a good exercise for them. Anyway, after discussing with them, we’ve come up with the following

Theorem 1 (Hon Leung-Ken’s theorem, 2011 (Cosine law for tetrahedron))

For a tetrahedron , the area of the triangle is given bywhere is the angle between and , is the angle between and , and is the angle between and .

*Proof:* We can w.l.o.g. assume that , then regarding the points as vectors in , we have

Thus

But

where is the normal of , and

we thus have

Remark 1Obviously the theorem can be generalized to an -dimensional simplex in , note also that the -dimensional volume of the simplex spanned by is(This also recovers the classical cosine law when . )

Google dihedral angle and Feng Luo, you may find very interesting versions Cosine Laws for Euclidean, Hyperbolic, and Spherical tetrahedra.

Thank you very much for your information. And indeed I’ve attended several of his lectures in the past (but it’s long time ago already so I’ve forgotten much about their details). Nevertheless I remember he’s talked about these cosine laws on simplices one or two times, I found it very interesting. ☺

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