truncated trapezohedron

{{Short description|Polyhedron made by cutting off a trapezohedron's polar vertices}}

{{Infobox polyhedron

| name = Set of {{nowrap|{{mvar|n}}-gonal}} truncated trapezohedra

| image = Pentagonal truncated trapezohedron.png

| caption = Example: pentagonal truncated trapezohedron (regular dodecahedron)

| euler =

| faces = 2 {{nowrap|{{mvar|n}}-sided}} polygons,
{{math|2n}} pentagons

| edges = {{math|6n}}

| vertices = {{math|4n}}

| vertex_config =

| schläfli =

| wythoff =

| coxeter =

| conway = {{math|1=[https://levskaya.github.io/polyhedronisme/?recipe=C100t4dA4 t4dA4]
[https://levskaya.github.io/polyhedronisme/?recipe=C100t5dA5 t5dA5]
[https://levskaya.github.io/polyhedronisme/?recipe=C100t6dA6 t6dA6]}}

| symmetry = {{math|Symmetry group#Three dimensions, [2{{sup|+}},2n], (2*n),}} order {{math|4n}}

| rotation_group = {{math|D{{sub|n}}, [2,n]{{sup|+}}, (22n),}} order {{math|2n}}

| surface_area =

| volume =

| dual = gyroelongated bipyramids

| properties = convex

| vertex_figure =

| net =

}}

In geometry, an {{nowrap|{{mvar|n}}-gonal}} truncated trapezohedron is a polyhedron formed by a {{nowrap|{{mvar|n}}-gonal}} trapezohedron with {{nowrap|{{mvar|n}}-gonal}} pyramids truncated from its two polar axis vertices.

The vertices exist as 4 {{nowrap|{{mvar|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.