The Cayley-Menger determinant is a determinant that gives the volume of a simplex
in 
dimensions. If 
and 
are integers with 
, 
is a 
-simplex in 
with vertices 
and 
denotes the 
matrix given by
 |
(1)
|
then the content 
is given by
 |
(2)
|
where 
is the 
matrix obtained from 
by bordering 
with a top row 
and a left column 
. Here, the vector l2-norms 
for 
are the edge lengths and the determinant in (2)
is the Cayley-Menger determinant (Sommerville 1958; Gritzmann and Klee 1994, p. 403).
The multiplicative inverses of the prefactors for 
, 1, 2, ... are 
, 2, 
, 288, 
, 460800, ... (OEIS A055546),
with the content of a 0-simplex taken to be 1.
For 
,
(2) becomes
 |
(3)
|
which gives the area for a plane triangle with side lengths 
,
,
and 
,
and is a form of Heron's formula.
For 
,
the content of the 3-simplex (i.e., volume of the general tetrahedron)
is given by the determinant
 |
(4)
|
where the edge between vertices 
and 
has length 
. Setting the left side equal to 0 (corresponding to a
tetrahedron of volume 0) gives a relationship between
the distances between vertices
of a planar quadrilateral (Uspensky 1948, p. 256).
Buchholz (1992, pp. 354-355) gives a slightly different (and slightly less symmetrical) form of this equation.
