Set of truncated trapezohedra
Pentagonal truncated trapezohedron
Faces2 n-gons,
2n pentagons
Symmetry groupDnd
Dual polyhedrongyroelongated dipyramids

An n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices.

The vertices exist as 4 n-agons in four parallel planes, with alternating orientation in the middle creating the pentagons.

The regular dodecahedron is the most common polyhedron in this class, being a platonic solid, and 12 congruent pentagonal faces.

A truncated trapezohedron has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramid, have all triangular faces. For example, the icosahedron is the dual of the dodecahedron.

Surface area

The surface area of a truncated trapezohedron whose base is a regular n sided polygon is therefore:

$ SA= 2(A_n + nA_5)= s^2(\frac{n}{2tan(\frac{180}{n})} + n\sqrt{\frac{25}{4} + \sqrt{\frac{125}{4}}}) $
$ SA= s^2(\frac{n}{(2n-4)tan(\frac{180}{n})} + \sqrt{\frac{225}{4}+ \sqrt{\frac{10125}{4}}}) $
$ SA= s^2(\frac{1}{2}sin(\frac{360}{n}) + \sqrt{\frac{225}{4}+ \sqrt{\frac{10125}{4}}})= s^2(\frac{1}{2csc(\frac{360}{n})} + \sqrt{\frac{225}{4}+ \sqrt{\frac{10125}{4}}}) $


Square truncated trapezohedron Pentagonal truncated trapezohedron Hexagonal truncated trapezohedron

External links

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