A novel algebraic relaxation employs a system of $18$ linear inequalities among the areas of the facets of a tetrahedron together with the areas of its three medial parallelograms. These tetrahedron inequalities are sufficiently powerful to ensure the 4-point Cayley-Menger determinant is non-negative in any metric space that also satisfies a quadratic trapazoid inequality. The seven areas in fact determine the tetrahedron up to isometry, although they also satisfy a simple quadratic identity and so cannot be treated as independent parameters.Next, I describe a geometric interpretation of the factors in Heron's formula as the lengths of the congruent pairs of segments into which the triangle's edges are divided by the ``in-touch points'' of its in-circle. Any three such non-negative numbers determine a triangle up to isometry, and I argue that makes them (not the edge lengths) the \underline{natural parameters} for triangle geometry. In tetrahedra, the analog of these natural parameters are the areas of the congruent pairs of triangles into which its facets are divided by the in-touch points of its in-sphere. They can be expressed as simple rational functions of the above seven areas, and conversely these areas are even simpler polynomial functions of the natural parameters. These results lead to a natural extension of Heron's formula to tetrahedra, and I will briefly describe how this formula extends beautifully to $n$-simplices for all $n \ge 2$. Finally, I discuss the singular nature of the zeros of the formula for a tetrahedron, which reveals a new class of degenerate Euclidean point configurations that are not coplanar but instead are separated by \underline{infinite} distances, with well-defined ratios, on a Euclidean line. The five-parameter set of such degenerate tetrahedra can be identified with a quotient of the Klein quadric by the action of a group of reflections isomorphic to $\mathbb Z_2^4$. A three-parameter subset thereof can be identified with quadruples of points in the ``special'' (area-preserving) affine plane, wherein all the distance ratios are indeterminate. As interesting as these observations may be, they would seem to indicate that converting bounds on distances into bounds on the areas, smoothing the areal bounds versus the tetrahedron inequalities, and then extracting tighter bounds on the distances from the results is not a viable approach to distance bound smoothing in general.