# Hypervolume of n-d simplex in an n+1 space

Hello,

This is my first time asking a question on this site so please let me know if I'm doing it wrong.

I have been trying to find out how to compute the hypervolume of an n-d simplex in an n+1 space.

I have found how to find the hypervolume of an n-d simplex in an n-d sapce, but don't know how to do it if the simplex is in an n+1 d space link

That link would show me how to compute the area of a triangle described in a 2-d space. I would need to know how to compute the area of a triangle described in a 3-d space.

Thanks guys

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I voted to close, but then could not refrain from answering, so should somehow retract the close vote... – Igor Rivin Jun 15 '12 at 20:48

I am a bit conflicted, since this is not a research level question. However, the answer is nice:

1. Move one of the vertices to the origin.
2. if the remaining vertices are now $v_1, \dots, v_n$ find, by solving a linear system, the vector $v_0$ orthogonal to all of the $v_i.$ Normalize so that the norm of $v_0$ equals $1.$
3. Find the volume of the simplex with vertices $0, v_0, \dots, v_n.$
4. Multiply by $n+1.$
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I did not know about the research level bit(I've now read the faq). Thanks for answering anyway. Would you mind telling me where I can look up why this works (I'm not a mathematician as I'm sure you've already guessed)? – edgar Jun 16 '12 at 4:20
It works, because the volume of the $n$-simplex is $1/n$ the height times the base. In your case the simplex in step 3 has your degenerate simplex as the base, and has height $1$ by the normalization... – Igor Rivin Jun 16 '12 at 4:59