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.

4Tetrahedrontriangular pyramid5 Square pyramid Triangular bipyramid6Pentagonal pyramidOctahedron (regular)triangular antiprismsquare bipyramidTriangular prismwedge7Hexagonal pyramidPentagonal bipyramidAugmented triangular prismElongated triangular pyramid8Heptagonal pyramidHexagonal bipyramidHexahedronCubeSquare prismCuboidRhombohedronTrigonal trapezohedronSquare antiprismTriakis tetrahedronElongated triangular bipyramidGyrobifastigiumSnub disphenoidBiaugmented triangular prism9Triangular cupolaTriaugmented triangular prismElongated square pyramidGyroelongated square pyramidTridiminished icosahedronOctagonal pyramidHeptagonal bipyramid10Nonagonal pyramidOctagonal bipyramidPentagonal prismPentagonal antiprismGyroelongated square bipyramidElongated square bipyramidMetabidiminished icosahedronAugmented tridiminished icosahedronSphenocorona11Decagonal pyramidNonagonal bipyramidAugmented pentagonal prismElongated pentagonal pyramidGyroelongated pentagonal pyramidAugmented sphenocoronaOctadecahedron12Elongated pentagonal bipyramidTruncated tetrahedronCuboctahedronSquare cupolaTriangular orthobicupolaBiaugmented pentagonal prismHendecagonal pyramidIcosahedronDecagonal bipyramidHexagonal prismHexagonal antiprismSphenomegacorona13Dodecagonal pyramidHendecagonal bipyramidAugmented hexagonal prism14Tridecagonal pyramidDodecagonal bipyramidElongated hexagonal dipyramidHeptagonal prismHeptagonal antiprismRhombic dodecahedronParabiaugmented hexagonal prismMetabiaugmented hexagonal prismHebesphenomegacoronaBilunabirotundaTetrakis hexahedron15Tetradecagonal pyramidTridecagonal bipyramidPentagonal cupolaElongated triangular cupolaGyroelongated triangular cupola16Octagonal prismOctagonal antiprismTriaugmented hexagonal prismAugmented truncated tetrahedronSquare orthobicupolaSquare gyrobicupolaDisphenocingulumSnub square antiprism17Hexadecagonal pyramidPentadecagonal bipyramid18Nonagonal prismEnneagonal antiprismGyroelongated triangular bicupolaElongated triangular orthobicupolaElongated triangular gyrobicupolarhombocuboctohedron1920DodecahedronPentagonal rotundaElongated square cupolaGyroelongated square cupolaPentagonal orthobicupolaPentagonal gyrobicupolaDecagonal prismDecagonal antiprism21Augmented dodecahedron22Parabiaugmented dodecahedronMetabiaugmented dodecahedron23Triaugmented dodecahedron24Truncated cubeTruncated octahedronRhombicuboctahedronElongated square gyrobicupolaGyroelongated square bicupolaSnub cube

External links

Counting Polyhedra