Set of truncated trapezohedra | |
---|---|
Faces | 2 n-gons, 2n pentagons |
Edges | 6n |
Vertices | 4n |
Symmetry group | D_{nd} |
Dual polyhedron | gyroelongated dipyramids |
Properties | convex |
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}}}) $
Forms
- Triangular truncated trapezohedron - 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid
- Square truncated trapezohedron - 8 pentagons, 2 squares, dual gyroelongated square dipyramid
- Pentagonal truncated trapezohedron or regular dodecahedron - 12 pentagonal faces, dual icosahedron
- Hexagonal truncated trapezohedron - 12 pentagons, 2 hexagons, dual gyroelongated hexagonal dipyramid
- ...
- n-agonal truncated trapezohedron - 2n pentagons, 2 n-agons, dual gyroelongated dipyramids
External links
- Conway Notation for Polyhedra Try: "tndAn", where n=4,5,6... example "t5dA5" is a dodecahedron.
This polyhedron-related article is a stub. You can help Math Wiki by expanding it. |
Community content is available under CC-BY-SA
unless otherwise noted.