# Truncated trapezohedron

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Set of truncated trapezohedra

Faces2 n-gons,
2n pentagons
Edges6n
Vertices4n
Symmetry groupDnd
Dual polyhedrongyroelongated dipyramids
Propertiesconvex

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:

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