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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.^{[citation needed]}

Set of n-gonal truncated trapezohedra | |
---|---|

Faces | 2 n-sided polygons, 2 n pentagons |

Edges | 6n |

Vertices | 4n |

Conway notation | t4dA4 t5dA5 t6dA6 |

Symmetry group | D_{nd}, [2^{+},2n], (2*n), order 4n |

Rotation group | D_{n}, [2,n]^{+}, (22n), order 2n |

Dual polyhedron | gyroelongated bipyramids |

Properties | convex |

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.

- 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**– 2*n*-gonal trapezohedron*n*pentagons, 2*n*-gons, dual gyroelongated dipyramids

- Conway Notation for Polyhedra Try: "t
**n**dA**n**", where**n**=4,5,6... example "t5dA5" is a dodecahedron.