List of small polyhedra by vertex count

In geometry, a polyhedron is a solid in three dimensions with flat faces and straight edges. Every edge has exactly two faces, and every vertex is surrounded by alternating faces and edges. The smallest polyhedron is the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices. Named polyhedra primarily come from the families of platonic solids, Archimedean solids, Catalan solids, and Johnson solids, as well as dihedral symmetry families including the pyramids, bipyramids, prisms, antiprisms, and trapezohedrons.

## Polyhedra by vertex count

Notes: Polyhedra with different names that are topologically identical are listed together. Except in the cases of four and five vertices, the lists below are by no means exhaustive of all possible polyhedra with the given number of vertices, but rather just include particularly simple/common/well-known/named examples. The "Counting Polyhedra" link below gives the exact number of distinct polyhedra with n vertices for small values of n.

4  |
Tetrahedron
triangular pyramid

|

5  |
Square pyramid |
Triangular bipyramid
6  |
Pentagonal pyramid |
Octahedron (regular)
triangular antiprism
square bipyramid  |
Triangular prism
wedge
7  |
Hexagonal pyramid |
Pentagonal bipyramid |
Augmented triangular prism |
Elongated triangular pyramid
8  | Heptagonal pyramid  |
Hexagonal bipyramid |
Hexahedron
Cube
Square prism
Cuboid
Rhombohedron
Trigonal trapezohedron |
Square antiprism |
Triakis tetrahedron |
Elongated triangular bipyramid

Gyrobifastigium |
Snub disphenoid |
Biaugmented triangular prism
9  |
Triangular cupola |
Triaugmented triangular prism |
Elongated square pyramid |
Gyroelongated square pyramid |
Tridiminished icosahedron
Octagonal pyramid  | Heptagonal bipyramid
10  | Nonagonal pyramid  |
Octagonal bipyramid |
Pentagonal prism |
Pentagonal antiprism |
Gyroelongated square bipyramid |
Elongated square bipyramid

Metabidiminished icosahedron |
Augmented tridiminished icosahedron |
Sphenocorona
11  | Decagonal pyramid  | Nonagonal bipyramid  |
Augmented pentagonal prism |
Elongated pentagonal pyramid |
Gyroelongated pentagonal pyramid |
Augmented sphenocorona

Octadecahedron
12  |
Elongated pentagonal bipyramid |
Truncated tetrahedron |
Cuboctahedron |
Square cupola |
Triangular orthobicupola |
Biaugmented pentagonal prism
Hendecagonal pyramid  |
Icosahedron |
Decagonal bipyramid |
Hexagonal prism |
Hexagonal antiprism |
Sphenomegacorona
13  | Dodecagonal pyramid  | Hendecagonal bipyramid  |
Augmented hexagonal prism
14  | Tridecagonal pyramid  | Dodecagonal bipyramid  |
Elongated hexagonal dipyramid  |
Heptagonal prism |
Heptagonal antiprism |
Rhombic dodecahedron

Parabiaugmented hexagonal prism |
Metabiaugmented hexagonal prism |
Hebesphenomegacorona |
Bilunabirotunda |
Tetrakis hexahedron
15  | Tetradecagonal pyramid  | Tridecagonal bipyramid  | Pentagonal cupola | Elongated triangular cupola
Gyroelongated triangular cupola
16  |
Octagonal prism |
Octagonal antiprism |
Triaugmented hexagonal prism |
Augmented truncated tetrahedron |
Square orthobicupola |
Square gyrobicupola

Disphenocingulum |
Snub square antiprism
17  | Hexadecagonal pyramid  | Pentadecagonal bipyramid
18  |
Nonagonal prism  |
Enneagonal antiprism | Gyroelongated triangular bicupola | Elongated triangular orthobicupola | Elongated triangular gyrobicupola | rhombocuboctohedron

20  |
Dodecahedron | Pentagonal rotunda | Elongated square cupola | Gyroelongated square cupola | Pentagonal orthobicupola | Pentagonal gyrobicupola

Decagonal prism |
Decagonal antiprism
21  | Augmented dodecahedron
22  | Parabiaugmented dodecahedron | Metabiaugmented dodecahedron
23  | Triaugmented dodecahedron
24  |
Truncated cube |
Truncated octahedron |
Rhombicuboctahedron |
Elongated square gyrobicupola |
Gyroelongated square bicupola

## External links

* Weisstein, Eric W.. "Polyhedron". http://mathworld.wolfram.com/Polyhedron.html.
* Counting Polyhedra

* v
* t
* e

Polyhedra

Listed by number of faces

1–10 faces|

* Monohedron
* Dihedron
* Trihedron
* Tetrahedron
* Pentahedron
* Hexahedron
* Heptahedron
* Octahedron
* Enneahedron
* Decahedron

11–20 faces|

* Hendecahedron
* Dodecahedron
* Tridecahedron
* Tetradecahedron
* Pentadecahedron
* Hexadecahedron
* Heptadecahedron
* Octadecahedron
* Enneadecahedron
* Icosahedron

Others|

* Triacontahedron (30)
* Hexecontahedron (60)
* Enneacontahedron (90)
* Hectotriadiohedron
* Apeirohedron

* v
* t
* e

Convex polyhedra

Platonic solids (regular)|

* tetrahedron
* cube
* octahedron
* dodecahedron
* icosahedron

Archimedean solids
(semiregular or uniform)

* truncated tetrahedron
* cuboctahedron
* truncated cube
* truncated octahedron
* rhombicuboctahedron
* truncated cuboctahedron
* snub cube
* icosidodecahedron
* truncated dodecahedron
* truncated icosahedron
* rhombicosidodecahedron
* truncated icosidodecahedron
* snub dodecahedron

Catalan solids
(duals of Archimedean)

* triakis tetrahedron
* rhombic dodecahedron
* triakis octahedron
* tetrakis hexahedron
* deltoidal icositetrahedron
* disdyakis dodecahedron
* pentagonal icositetrahedron
* rhombic triacontahedron
* triakis icosahedron
* pentakis dodecahedron
* deltoidal hexecontahedron
* disdyakis triacontahedron
* pentagonal hexecontahedron

Dihedral regular|

* dihedron
* hosohedron

Dihedral uniform|

* prisms
* antiprisms

duals:|

* bipyramids
* trapezohedra

Dihedral others|

* pyramids
* truncated trapezohedra
* gyroelongated bipyramid
* cupola
* bicupola
* pyramidal frusta
* bifrustum
* rotunda
* birotunda
* prismatoid
* scutoid

Degenerate polyhedra are in italics.

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Original source: https://en.wikipedia.org/wiki/List of small polyhedra by vertex count. Read more

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