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	File:Tetrahedron.svg Tetrahedron triangular pyramid
5	File:Square pyramid.png Square pyramid	File:Triangular dipyramid.png Triangular bipyramid
6	File:Pentagonal pyramid.png Pentagonal pyramid	File:Octahedron.svg Octahedron (regular) triangular antiprism square bipyramid	File:Triangular prism.png Triangular prism wedge
7	File:Hexagonal pyramid.png Hexagonal pyramid	File:Pentagonal dipyramid.png Pentagonal bipyramid	File:Augmented triangular prism.png Augmented triangular prism	File:Elongated triangular pyramid.png Elongated triangular pyramid
8	Heptagonal pyramid	File:Hexagonale bipiramide.png Hexagonal bipyramid	File:Hexahedron.png Hexahedron Cube Square prism Cuboid Rhombohedron Trigonal trapezohedron	File:Square antiprism.png Square antiprism	File:Triakistetrahedron.jpg Triakis tetrahedron	File:Elongated triangular dipyramid.png Elongated triangular bipyramid
	File:Gyrobifastigium.png Gyrobifastigium	File:Snub disphenoid.png Snub disphenoid	File:Biaugmented triangular prism.png Biaugmented triangular prism
9	File:Triangular cupola.png Triangular cupola	File:Triaugmented triangular prism.png Triaugmented triangular prism	File:Elongated square pyramid.png Elongated square pyramid	File:Gyroelongated square pyramid.png Gyroelongated square pyramid	File:Tridiminished icosahedron.png Tridiminished icosahedron
	Octagonal pyramid	Heptagonal bipyramid
10	Nonagonal pyramid	File:Octagonal bipyramid.png Octagonal bipyramid	File:Pentagonal prism.png Pentagonal prism	File:Pentagonal antiprism.png Pentagonal antiprism	File:Gyroelongated square dipyramid.png Gyroelongated square bipyramid	File:Elongated square dipyramid.png Elongated square bipyramid
	File:Metabidiminished icosahedron.png Metabidiminished icosahedron	File:Augmented tridiminished icosahedron.png Augmented tridiminished icosahedron	File:Sphenocorona.png Sphenocorona
11	Decagonal pyramid	Nonagonal bipyramid	File:Augmented pentagonal prism.png Augmented pentagonal prism	File:Elongated pentagonal pyramid.png Elongated pentagonal pyramid	File:Gyroelongated pentagonal pyramid.png Gyroelongated pentagonal pyramid	File:Augmented sphenocorona.png Augmented sphenocorona
	File:Octadecahedron.png Octadecahedron
12	File:Elongated pentagonal dipyramid.png Elongated pentagonal bipyramid	File:Truncated tetrahedron.png Truncated tetrahedron	File:Cuboctahedron.png Cuboctahedron	File:Square cupola.png Square cupola	File:Triangular orthobicupola.png Triangular orthobicupola	File:Biaugmented pentagonal prism.png Biaugmented pentagonal prism
	Hendecagonal pyramid	File:Icosahedron.svg Icosahedron	File:Decagonal bipyramid.png Decagonal bipyramid	File:Hexagonal prism.png Hexagonal prism	File:Hexagonal antiprism.png Hexagonal antiprism	File:Sphenomegacorona.png Sphenomegacorona
13	Dodecagonal pyramid	Hendecagonal bipyramid	File:Augmented hexagonal prism.png Augmented hexagonal prism
14	Tridecagonal pyramid	Dodecagonal bipyramid	File:Elongated hexagonal dipyramid.png Elongated hexagonal dipyramid	File:Prism 7.png Heptagonal prism	File:Antiprism 7.png Heptagonal antiprism	File:Rhombic dodecahedron.png Rhombic dodecahedron
	File:Parabiaugmented hexagonal prism.png Parabiaugmented hexagonal prism	File:Metabiaugmented hexagonal prism.png Metabiaugmented hexagonal prism	File:Hebesphenomegacorona.png Hebesphenomegacorona	File:Bilunabirotunda.png Bilunabirotunda	File:Tetrakishexahedron.jpg Tetrakis hexahedron
15	Tetradecagonal pyramid	Tridecagonal bipyramid	Pentagonal cupola	Elongated triangular cupola
	Gyroelongated triangular cupola
16	File:Octagonal prism.png Octagonal prism	File:Octagonal antiprism.png Octagonal antiprism	File:Triaugmented hexagonal prism.png Triaugmented hexagonal prism	File:Augmented truncated tetrahedron.png Augmented truncated tetrahedron	File:Square orthobicupola.png Square orthobicupola	File:Square gyrobicupola.png Square gyrobicupola
	File:Disphenocingulum.png Disphenocingulum	File:Snub square antiprism.png Snub square antiprism
17	Hexadecagonal pyramid	Pentadecagonal bipyramid
18	File:Prism 9.png Nonagonal prism	File:Enneagonal antiprism.png Enneagonal antiprism	Gyroelongated triangular bicupola	Elongated triangular orthobicupola	Elongated triangular gyrobicupola	rhombocuboctohedron
19
20	File:Dodecahedron.svg Dodecahedron	Pentagonal rotunda	Elongated square cupola	Gyroelongated square cupola	Pentagonal orthobicupola	Pentagonal gyrobicupola
	File:Decagonal prism.png Decagonal prism	File:Decagonal antiprism.png Decagonal antiprism
21	Augmented dodecahedron
22	Parabiaugmented dodecahedron	Metabiaugmented dodecahedron
23	Triaugmented dodecahedron
24	File:Truncated hexahedron.png Truncated cube	File:Truncated octahedron.png Truncated octahedron	File:Small rhombicuboctahedron.png Rhombicuboctahedron	File:Pseudorhombicuboctahedron.png Elongated square gyrobicupola	File:Gyroelongated square bicupola.png Gyroelongated square bicupola	File:Snubhexahedroncw.jpg Snub cube

External links

• Weisstein, Eric W. "Polyhedron". MathWorld.
• Counting Polyhedra