| Regular triacontagon | |
|---|---|
 A regular triacontagon | |
| Type | Regular polygon |
| Edges and vertices | 30 |
| Schläfli symbol | {30}, t{15} |
| Coxeter–Dynkin diagrams |      |
| Symmetry group | Dihedral (D30), order 2×30 |
| Internal angle (degrees) | 168° |
| Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
| Dual polygon | Self |
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
The regular triacontagon is a constructible polygon, by an edge-bisection of a regular pentadecagon, and can also be constructed as a truncated pentadecagon, t{15}. A truncated triacontagon, t{30}, is a hexacontagon, {60}.
One interior angle in a regular triacontagon is 168 degrees, meaning that one exterior angle would be 12°. The triacontagon is the largest regular polygon whose interior angle is the sum of the interior angles of smaller polygons: 168° is the sum of the interior angles of the equilateral triangle (60°) and the regular pentagon (108°).
The area of a regular triacontagon is (with t = edge length) [1]
The inradius of a regular triacontagon is
The circumradius of a regular triacontagon is
As 30 = 2 × 3 × 5, a regular triacontagon is constructible using a compass and straightedge. [2]
The regular triacontagon has Dih30 dihedral symmetry, order 60, represented by 30 lines of reflection. Dih30 has 7 dihedral subgroups: Dih15, (Dih10, Dih5), (Dih6, Dih3), and (Dih2, Dih1). It also has eight more cyclic symmetries as subgroups: (Z30, Z15), (Z10, Z5), (Z6, Z3), and (Z2, Z1), with Zn representing π/n radian rotational symmetry.
John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. [3] He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular triacontagons. Only the g30 subgroup has no degrees of freedom but can be seen as directed edges.
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. [4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontagon, m=15, it can be divided into 105: 7 sets of 15 rhombs. This decomposition is based on a Petrie polygon projection of a 15-cube.
 |  |  |  |  |
A triacontagram is a 30-sided star polygon (though the word is extremely rare). There are 3 regular forms given by Schläfli symbols {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same vertex configuration.
| Compounds and stars | |||||||
|---|---|---|---|---|---|---|---|
| Form | Compounds | Star polygon | Compound | ||||
| Picture |  {30/2}=2{15} |  {30/3}=3{10} |  {30/4}=2{15/2} |  {30/5}=5{6} |  {30/6}=6{5} |  {30/7} |  {30/8}=2{15/4} |
| Interior angle | 156° | 144° | 132° | 120° | 108° | 96° | 84° |
| Form | Compounds | Star polygon | Compound | Star polygon | Compounds | ||
| Picture |  {30/9}=3{10/3} |  {30/10}=10{3} |  {30/11} |  {30/12}=6{5/2} |  {30/13} |  {30/14}=2{15/7} |  {30/15}=15{2} |
| Interior angle | 72° | 60° | 48° | 36° | 24° | 12° | 0° |
There are also isogonal triacontagrams constructed as deeper truncations of the regular pentadecagon {15} and pentadecagram {15/7}, and inverted pentadecagrams {15/11}, and {15/13}. Other truncations form double coverings: t{15/14}={30/14}=2{15/7}, t{15/8}={30/8}=2{15/4}, t{15/4}={30/4}=2{15/4}, and t{15/2}={30/2}=2{15}. [5]
| Compounds and stars | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Quasiregular | Isogonal | Quasiregular Double coverings | |||||||||
 t{15} = {30} |  |  |  |  |  |  |  |  t{15/14}=2{15/7} | |||
 t{15/7}={30/7} |  |  |  |  |  |  |  |  t{15/8}=2{15/4} | |||
 t{15/11}={30/11} |  |  |  |  |  |  |  |  t{15/4}=2{15/2} | |||
 t{15/13}={30/13} |  |  |  |  |  |  |  |  t{15/2}=2{15} | |||
The regular triacontagon is the Petrie polygon for three 8-dimensional polytopes with E8 symmetry, shown in orthogonal projections in the E8 Coxeter plane. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H4 Coxeter plane.
| E8 | H4 | |||
|---|---|---|---|---|
 421 |  241 |  142 |  120-cell |  600-cell |
The regular triacontagram {30/7} is also the Petrie polygon for the great grand stellated 120-cell and grand 600-cell.
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.
In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon.
In geometry, an octagon is an eight-sided polygon or 8-gon.
In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.
In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.
In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non–self-intersecting 65537-gon is 11796300°.
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.
In mathematics, a hexadecagon is a sixteen-sided polygon.
In geometry, an octadecagon or 18-gon is an eighteen-sided polygon.
A megagon or 1,000,000-gon (million-gon) is a polygon with one million sides.
In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
