Truncated Cube


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The 14-faced Archimedean solid with faces 8{3}+6{8}. It is also the uniform polyhedron with Maeder index 9 (Maeder 1997), Wenninger index 8 (Wenninger 1989), Coxeter index 21 (Coxeter et al. 1954), and Har'El index 14 (Har'El 1993). It has Schläfli symbol t{4,3} and Wythoff symbol 23|4. It is illustrated above together with a wireframe version and a net that can be used for its construction.

It is implemented in the Wolfram Language as PolyhedronData["TruncatedCube"] or UniformPolyhedron["TruncatedCube"]. Precomputed properties are available as PolyhedronData["TruncatedCube", prop].


The truncated cube is the convex hull of the great cubicuboctahedron, great rhombihexahedron, and quasirhombicuboctahedron uniform polyhedra.


The dual polyhedron of the truncated cube is the small triakis octahedron, both of which are illustrated above together with their common midsphere. The inradius r of the dual, midradius rho of the solid and dual, and circumradius R of the solid for a=1 are

r=1/(17)(5+2sqrt(2))sqrt(7+4sqrt(2)) approx 1.63828
rho=1/2(2+sqrt(2)) approx 1.70711
R=1/2sqrt(7+4sqrt(2)) approx 1.77882.

The distances from the center of the solid to the centroids of the triangular and octagonal faces are


The surface area and volume are


The unit truncated cube has Dehn invariant


where the first expression uses the basis of Conway et al. (1999).

See also

Archimedean Solid, Equilateral Zonohedron, Icositetrahedron, Truncated Cubical Graph, Truncation

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 138, 1987.Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Cundy, H. and Rollett, A. "Truncated Cube. 3.8^2." §3.7.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 103, 1989.Geometry Technologies. "Truncated Cube."'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kasahara, K. "Two New Semiregular Polyhedrons." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 227, 1988.Maeder, R. E. "09: Truncated Cube." 1997., M. J. "The Truncated Hexahedron (Cube)." Model 8 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 22, 1989.

Cite this as:

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

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