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What is Heron's formula in n dimensions?
In plane geometry, knowing all the side lengths of a polygon of m sides isn't enough, by itself, to determine the area; unless m=3, in which case there is Heron's formula:
A = √[s(s - a)(s - b)(s - c)], where s (the semiperimeter) = ½(a + b + c)
In solid geometry, a similar situation exists for the tetrahedron -- knowing all 6 edge lengths is sufficient to completely determine its shape, and thus, its volume.
A) What is the Heron-like formula for that?
B) What is the formula for a simplex (hyperpyramid) in n dimensions, given all ½n(n+1) edge lengths?
Unlike my first Y!A question, I don't have prior knowledge of the answer to this.
And elegance/simplicity in the final expression will get extra consideration.
Don't think you have to have the full n-dimensional solution before speaking up!
Can you give the formula for n=3? Or even some insight into how to go about getting it? If you have a useful contribution, by all means, let's hear it!
Correction: In 3D, knowing all 6 edges, you still need to know which pairs are opposite each other, because the volume is in general, different when the same 6 lengths are assembled differently. Similarly, in higher dimensions, it will matter how the ½n(n+1) edges are connected.
I'm having second thoughts about the general question, although the 3D case is still valid.