A hexahedron is a polyhedron with six faces. The figure above shows a number of named hexahedra, in particular the acute golden rhombohedron, cube, cuboid, hemicube, hemiobelisk, obtuse golden rhombohedron, pentagonal pyramid, pentagonal wedge, tetragonal antiwedge, and triangular dipyramid.


There are seven topologically distinct convex hexahedra, corresponding through graph duality with the seven hexahedral graphs. The illustration above shows these seven hexahedra (top line), their skeletons (middle line), and the hexahedral graphs whose duals correspond to the polyhedra and their skeletons (bottom line).

The unique regular hexahedron is the cube and the unique chiral hexahedron is the tetragonal antiwedge.

Two hexahedra can be built from regular polygons with equal edge lengths: the equilateral triangular dipyramid and pentagonal pyramid. Rhombohedra are a special class of hexahedron in which opposite faces are congruent rhombi.

Through graph duality, the list of numbers of vertices for each polyhedron in a hexahedron corresponds to the degree sequence (sequence of vertex degrees) of a hexahedral graph. The following table lists the hexahedra, together with their degree sequences, number of vertices V, and number of edges E, which are related through the polyhedral formula. Standard names do not appear to be in common use for a number of these; for such cases, the names appearing on Michon are used.

hexahedrondegree sequenceVE
triagular dipyramid(3, 3, 3, 3, 3, 3)59
pentagonal pyramid(3, 3, 3, 3, 3, 5)610
tetragonal antiwedge(3, 3, 3, 3, 4, 4)610
hemiobelisk(3, 3, 3, 4, 4, 5)711
hemicube(3, 3, 4, 4, 4, 4)711
pentagonal wedge(3, 3, 4, 4, 5, 5)812
cube(4, 4, 4, 4, 4, 4)812

See also

Cube, Cuboid, Hemicube, Hemiobelisk, Hexagonal Pyramid, Hexahedral Graph, Pentagonal Wedge, Polyhedron, Rhombohedron, Tetragonal Antiwedge, Triangular Dipyramid

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Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (3-Connected Planar) Graphs." Math. Comput. 37, 523-532, 1981.Gardner, M. "Find the Hexahedrons." §19.9 in Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 224-225 and 233, 1966.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.McClellan, J. "The Hexahedra Problem." Recr. Math. Mag., No. 4, 34-40, Aug. 1961.Michon, G. P. "Final Answers: Polyhedra & Polytopes.", J. "Problème de situation." Ann. de Math 19, 36, 1828. Reprinted in Jacob Steiner's gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 227, 1971.

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Cite this as:

Weisstein, Eric W. "Hexahedron." From MathWorld--A Wolfram Web Resource.

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