  We introduce the n-th Heron variety as the realization space of the (squared)
volumes of faces of an n-simplex. Our primary goal is to understand the extent
to which Heron's formula, which expresses the area of a triangle as a function
of its three edge lengths, can be generalized. Such a formula for one face
volume of an n-simplex in terms of other face volumes expresses a dependence in
the algebraic matroid of the Heron variety. Whether the volume is expressible
in terms of radicals is controlled by the monodromy groups of the coordinate
projections of the Heron variety onto coordinates of bases. We discuss a suite
of algorithms, some new, for determining these matroids and monodromy groups.
We apply these algorithms toward the smaller Heron varieties, organize our
findings, and interpret the results in the context of our original motivation.
