Is there a generalized version of Heron's formula for calculating the equivalent of a "volume" of an n-dimensional "n-angle" based on the length of it's sides? I've seen the equivalent formula for a tetrahedron, but I'd like to keep extending the shape by adding an extra point that connects to all existing points in the next dimension.

Does that make sense?

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    $\begingroup$ n-angle $\mapsto$ simplex $\:$ ? $\;\;\;$ $\endgroup$ – user57159 Oct 9 '13 at 5:11

Yes. See Cayley-Menger Determinant


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