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# Truncated Dodecahedron

## Make Your Own Truncated Dodecahedron

The truncated dodecahedron is the 32-faced Archimedean solid with faces . It is also uniform polyhedron with Maeder index 26 (Maeder 1997), Wenninger index 10 (Wenninger 1989), Coxeter index 29 (Coxeter et al. 1954), and Har'El index 31 (Har'El 1993). It has Schläfli symbol t and Wythoff symbol . It is illustrated above together with a wireframe version and a net that can be used for its construction.

Some symmetric projections of the truncated dodecahedron are illustrated above.

It is implemented in the Wolfram Language as UniformPolyhedron["TruncatedDodecahedron"] or PolyhedronData["TruncatedDodecahedron"].

The truncated dodecahedron is the convex hull of the great ditrigonal dodecicosidodecahedron, great dodecicosahedron, and great icosicosidodecahedron uniform polyhedra.

To construct the truncated dodecahedron by truncation, note that we want the inradius of the truncated pentagon to correspond with that of the original pentagon, , of unit side length . This means that the side lengths of the decagonal faces in the truncated dodecahedron satisfy

 (1)

giving

 (2)

The length of the corner which is chopped off is therefore given by

 (3)

The dual polyhedron of the truncated dodecahedron is the triakis icosahedron, both of which are illustrated above together with their common midsphere. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are

 (4) (5) (6)

The distances from the center of the solid to the centroids of the triangular and decagonal faces are given by

 (7) (8)

The surface area and volume are

 (9) (10)

The unit truncated dodecahedron has Dehn invariant

 (11) (12)

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

Archimedean Solid, Equilateral Zonohedron, Hexecontahedron, Triakis Icosahedron, Truncated Dodecahedron-Triakis Icosahedron Compound, Truncation

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## References

Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.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 Dodecahedron. ." §3.7.9 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 109, 1989.Geometry Technologies. "Truncated Dodecahedron." http://www.scienceu.com/geometry/facts/solids/tr_dodeca.html.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kasahara, K. "The Final Semiregular Polyhedron." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 229, 1988.Maeder, R. E. "26: Truncated Dodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/26.html.Wenninger, M. J. "The Truncated Dodecahedron." Model 10 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 24, 1989.

## Cite this as:

Weisstein, Eric W. "Truncated Dodecahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TruncatedDodecahedron.html