Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site ho95b.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!mhuxt!houxm!ho95b!ran
From: ran@ho95b.UUCP (RANeinast)
Newsgroups: net.math
Subject: Re: volume of a tetrahedron
Message-ID: <421@ho95b.UUCP>
Date: Fri, 24-May-85 08:35:36 EDT
Article-I.D.: ho95b.421
Posted: Fri May 24 08:35:36 1985
Date-Received: Sat, 25-May-85 08:38:59 EDT
Organization: AT&T-Bell Labs, Holmdel, NJ
Lines: 48

>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


Here's the expression, in the easiest form to remember it:

(1/12)*sqrt( a^2*d^2*(-a^2+b^2+c^2-d^2+e^2+f^2)
             b^2*e^2*(+a^2-b^2+c^2+d^2-e^2+f^2)
             c^2*f^2*(+a^2+b^2-c^2+d^2+e^2-f^2)
            -a^2*b^2*f^2-a^2*c^2*e^2-b^2*c^2*d^2-d^2*e^2*f^2).

Here's how the tetrahedron's edges are lettered:


     |`\
     |` \
     | ` \
     | `  \
     |  `  \
    c|  `d  \e
     |   `   \
     |   /`   \
     | b/  ``  \
     | /    f`` \
     |/        ``\
     -------------
          a

The secret is that (a, d), (b, e), and (c, f) are opposite, non-touching
edges.  Keeping that straight helps with the first 3 lines in the formula
(note where the minus signs are).  The last 4 subtracted terms are just
made up of the triangular faces of the tetrahedron.

I had played around with this formula in high school, and it took
me about 10 minutes to rederive it (I know a lot more fancy tricks now).
No, there is no simple factorization as there is for Hero's formula
(unless you have some special case like a=d, b=e, c=f [in fact, you can
use the special case factoriation to show that a general case factorization
does not exist]).
-- 

". . . and shun the frumious Bandersnatch."
Robert Neinast (ihnp4!ho95b!ran)
AT&T-Bell Labs