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From: igor@fisher.UUCP (Igor Rivin)
Newsgroups: net.math
Subject: Re: volume of a tetrahedron
Message-ID: <629@fisher.UUCP>
Date: Sat, 25-May-85 15:51:37 EDT
Article-I.D.: fisher.629
Posted: Sat May 25 15:51:37 1985
Date-Received: Sun, 26-May-85 01:13:41 EDT
References: <758@gloria.UUCP>
Distribution: net
Organization: Princeton University Department of Statistics
Lines: 29

> Does anybody know of a _compact_ expression for the volume
> of a tetrahedron in terms of its edges--something comparable
> to Hero's and Brahmagupta's formulas in plane geometry?  I
> have an expression but it's rather long and hard to remember.
> -- 
> Col. G. L. Sicherman
> ...{rocksvax|decvax}!sunybcs!colonel

*** REPLACE THIS LINE WITH YOUR MESSAGE ***

Yes, there is a formula. Let Aij be the length of the side connecting the 
ith and the jth vertices. Than V^2 =
                   
			| 0	1	1	1	1     |
			|				      |
			| 1	0	A12^2   A13^2   A14^2 |		 
			|				      |	
	1/(288) *	| 1	A12^2	0       A23^2   A24^2 |		 
			|				      |
			| 1     A13^2	A23^2	0	A34^2 | 
			|				      |
			| 1	A14^2	A24^2	A34^2	0     |

This formula (with the obvious generalisations) works for all dimensions.

The constant, in general, is (-1)^(n-1)/(2^n*(n!)^2), in n dimensions.


Igor.