Some Problems About Tetrahedra

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York


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It is not easy given six stick lengths to tell if there is a tetrahedron with these lengths. However, there are, in fact, ways to answer this question. In particular if there are six different stick lengths it is known that there can be no more than 30 tetrahedra with these particular lengths.

Here are a few research questions centered around this circle of ideas.

Question 1:

Given that six different lengths will determine 30 different tetrahedra is there a way of telling which of the 30 tetrahedra has the smallest volume and which one has the largest volume by how the 6 edges are positioned with respect to each other?

Question 2:

It is known that six distinct lengths can determine either 0 or 30 different tetrahedra. What other numbers between 0 and 30 can occur for the number of tetrahedra with six distinct edge lengths?


Malkevitch, J., and D. Mussa, The transition from two dimensions to three dimensions- some geometry of the tetrahedron, Consortium Number 105, Fall/Winter 2013, p. 1-5.