  A natural extension of Heron's 2000 year old formula for the area of a
triangle to the volume of a tetrahedron is presented. This gives the fourth
power of the volume as a polynomial in six simple rational functions of the
areas of its four faces and three medial parallelograms, which will be referred
to herein as "interior faces." Geometrically, these rational functions are the
areas of the triangles into which the exterior faces are divided by the points
at which the tetrahedron's in-sphere touches those faces. Part I presents an
overview of these results and some necessary but little-known background in
areal geometry. Part II derives the promised extension, and ends with a
conjecture as to how the formula extends to $n$-dimensional simplices for all
$n > 3$. Part III explains how, for $n = 3$, the zeros of the polynomial
constitute a five-dimensional semi-algebraic variety consisting almost entirely
of collinear tetrahedra with vertices separated by infinite distances, but with
generically well-defined distance ratios; it further proves that these
unconventional Euclidean configurations can be identified with a quotient of
the Klein quadric by an action of a group of reflections isomorphic to $\mathbb
Z_2^4$, wherein four-point configurations in the affine plane constitute a
distinguished three-dimensional subvariety. Part IV consists of five appendices
which show, among other things, that the algebraic structure of the zeros in
the affine plane naturally defines the associated four-element, rank $3$
chirotope, aka affine oriented matroid.
