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also see Defining Geometric Figures

  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.

Let   dij, 0 < i < j < 5   be the length of the edge connecting vertices i and j.

Let V be the volume of the tetrahedron. Then

    288 V2 =

     0 

     d122 

     d132 

     d142 

     1 


     d122 

     0 

     d232 

     d242 

     1 

     d132 

     d232 

     0 

     d342 

     1 

     d142 

     d242 

     d342 

     0 

     1 

     1 

     1 

     1 

     1 

     0 

 
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.

Contributed by "Dr. Rob," Robert L. Ward

Back to Regular Tetrahedron

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