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I am looking for a general formula to calculate the volume of a euclidean simplex in any number of dimensions. On Wikipedia I found that a formula similar to Heron's formula can be applied to tetrahedrons as well as triangles, can this perhaps be abstracted to a formula for the volume of an n-simplex? or perhaps another type of formula?

(Wikipedia page: http://en.wikipedia.org/wiki/Heron's_formula#Heron-type_formula_for_the_volume_of_a_tetrahedron)

Ushwald
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1 Answers1

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See http://mathworld.wolfram.com/Cayley-MengerDeterminant.html. Given an $n$-simplex with vertices $v_i$, put $B_{ij}=\|v_i-v_j\|^2$ for $0\leq i,j\leq n$. Then put $B_{n+1,j}=B_{i,n+1}=1$ except $B_{n+1,n+1}=0$. The volume is then $$ V = \sqrt{(-1)^{n+1}\frac{\det(B)}{2^nn!^2}} $$

Neil Strickland
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