Pentakis Dodecahedron


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The pentakis dodecahedron is the 60-faced dual polyhedron of the truncated icosahedron (Holden 1971, p. 55). It can be constructed by augmentation of a unit edge-length dodecahedron by a pyramid with height 1/(19)sqrt(1/5(65+22sqrt(5))). It is illustrated above together with a wireframe version and a net that can be used for its construction.

It is Wenninger dual W_9.

Solids inscriptable in a pentakis dodecahedron

A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the pentakis dodecahedron (E. Weisstein, Dec. 25-27, 2009).


The pentakis dodecahedron is he convex hull of the small triambic icosahedron hull.

Taking the dual of a truncated icosahedron with unit edge lengths gives a pentakis dodecahedron with edge lengths


Normalizing so that s_1=1, the surface area and volume are


See also

Archimedean Dual, Archimedean Solid, Dual Polyhedron, Hexecontahedron, Pentakis Dodecahedral Graph, Truncated Icosahedron

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Holden, A. Shapes, Space, and Symmetry. New York: Columbia University Press, p. 55, 1971.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 153, 2002.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 18, 1983.

Cite this as:

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

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