Heron in space

As with a triangle in the plane, a tetrahedron is completely determined by the lengths of its six edges. So is there an analogue of Heron's formula for a tetrahedron?

Yes. The great Swiss mathematician Leonhard Euler (1707–1783), apparently drawing on Tartaglia's work during the Renaissance, discovered such a formula for a tetrahedron. In this formula, a, b, c, d, e and f are the edge lengths of the tetrahedron, with edge a opposite e, edge c opposite d, and edge b opposite f. The volume of the tetrahedron—the three-dimensional analogue of a triangle's area—is denoted V. Thus:

12² V² = (a ²e ² + b² f ² + c ²d ²)(a ² + b ² + c ² + d ² + e ² + f ²) – 2(a ²e ² (a ² + e ²) \+ b ²f ²(b ² + f ²)+ c ²d ²(c ² + d ²)) – (a ²b ²d ² + a ²c ²f ² + b ²c ²e ² + d ²e ²f ²).