  For n greater than or equal to 4, the square of the volume of an n-simplex
satisfies a polynomial relation with coefficients depending on the squares of
the areas of 2-faces of this simplex. First, we compute the minimal degree of
such polynomial relation. Second, we prove that the volume an n-simplex
satisfies a monic polynomial relation with coefficients depending on the areas
of 2-faces of this simplex if and only if n is even and at least 6, and we
study the leading coefficients of polynomial relations satisfied by the volume
for other n.
