title: Truncated trapezohedron
text: # Truncated trapezohedron
  
  In geometry, an n-gonal truncated trapezohedron is a polyhedron formed by a n-gonal trapezohedron with n-gonal pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism.
  
  The vertices exist as 4 n-gons 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, with 12 congruent pentagonal faces.
  
  A truncated trapezohedron has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramids, have all triangular faces. For example, the icosahedron is the dual of the dodecahedron.
  
  ## Forms
  
  * Triangular truncated trapezohedron (Dürer's solid) – 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid
  * Truncated square trapezohedron – 8 pentagons, 2 squares, dual gyroelongated square dipyramid
  * Truncated pentagonal trapezohedron or regular dodecahedron – 12 pentagonal faces, dual icosahedron
  * Truncated hexagonal trapezohedron – 12 pentagons, 2 hexagons, dual gyroelongated hexagonal dipyramid
  * Truncated n-gonal trapezohedron – 2n pentagons, 2 n-gons, dual gyroelongated dipyramids
  