TITLE: Generalized Heron's formula for n-dimensional "n-angle" instead of "triangle"
QUESTION [8 upvotes]: 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?

REPLY [11 votes]: Yes.  See  Cayley-Menger Determinant