Polyhedra: Formulas
Irregular Tetrahedron


A tetrahedron is a polyhedron with four planar faces (each of which is a triangle), six edges, and four vertices. It is irregular if and only if the faces are not all equilateral triangles, which happens if and only if the edges do not all have the same lengths, which happens if and only if the face angles are not all of equal measure. The following method will find the volume of any tetrahedron, but there is a simpler formula if the tetrahedron is regular. Number the vertices of the tetrahedron 1, 2, 3, and 4.
 
Of course the distances must obey the triangle inequality for each face, and in addition, the value of the above determinant must be positive, in order for them to be the edge lengths of a tetrahedron. For example, the numbers {4,4,4,4,4,7} obey the triangle inequalities, but the value of the above determinant is negative (1568), and no tetrahedron has edges with those lengths. This formula is given in J. V. Uspensky, The Theory of Equations (1948). Its original discoverer is unknown to me. 
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