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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 . It is illustrated above together with a wireframe version and a net that can be used for its construction.

It is Wenninger dual .

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 , the surface area and volume are

The inscribed realization illustrated above has the same combinatorial structure as the dual polyhedron described earlier, but different proportions. The dual polyhedron has vertices at two distances from its center, whereas all 32 vertices of the inscribed realization lie on one sphere. It is obtained from the convex hull of the 12 vertices of a regular icosahedron on the unit sphere together with the radial projections of its 20 face centers onto the sphere. Hoehner (2026) proved that this realization uniquely maximizes volume among convex polyhedra with at most 32 vertices on the unit sphere, up to orthogonal transformations. Its volume is .

See also

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

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References

Hoehner, S. "On Maximum Volume Polytopes Inscribed in the Sphere." 7 Nov 2026. https://arxiv.org/abs/2611.10777.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.

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Pentakis Dodecahedron

Cite this as:

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

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