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List of small polyhedra by vertex count

From Wikipedia, the free encyclopedia

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

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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
19
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

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