| Regular triacontagon | |
|---|---|
 A regular triacontagon | |
| Type | Regular polygon |
| Edges andvertices | 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 |
Ingeometry, atriacontagon or 30-gon is a thirty-sidedpolygon. The sum of any triacontagon's interior angles is5040 degrees.
Theregular triacontagon is aconstructible polygon, by an edge-bisection of a regularpentadecagon, and can also be constructed as atruncatedpentadecagon, t{15}. Atruncated triacontagon, t{30}, is ahexacontagon, {60}.
One interior angle in aregular 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 theinterior angles of smallerpolygons: 168° is the sum of the interior angles of theequilateral triangle (60°) and theregular pentagon (108°).
Thearea of a regular triacontagon is (witht = edge length)[1]
Theinradius of a regular triacontagon is
Thecircumradius of a regular triacontagon is
As 30 = 2 × 3 × 5 , a regular triacontagon isconstructible using acompass and straightedge.[2]
Theregular triacontagon has Dih30dihedral 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 morecyclic 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 givesd (diagonal) with mirror lines through vertices,p with mirror lines through edges (perpendicular),i with mirror lines through both vertices and edges, andg for rotational symmetry.a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular triacontagons. Only theg30 subgroup has no degrees of freedom but can be seen asdirected edges.
Coxeter states that everyzonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected intom(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 theregular triacontagon,m=15, it can be divided into 105: 7 sets of 15 rhombs. This decomposition is based on aPetrie polygon projection of a15-cube.
 |  |  |  |  |
A triacontagram is a 30-sidedstar polygon (though the word is extremely rare). There are 3 regular forms given bySchläfli symbols {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the samevertex configuration.
| Compounds and stars | |||||||
|---|---|---|---|---|---|---|---|
| Form | Compounds | Star polygon | Compound | ||||
| Picture | .svg) {30/2}=2{15} | .svg) {30/3}=3{10} | .svg) {30/4}=2{15/2} | .svg) {30/5}=5{6} | .svg) {30/6}=6{5} |  {30/7} | .svg) {30/8}=2{15/4} |
| Interior angle | 156° | 144° | 132° | 120° | 108° | 96° | 84° |
| Form | Compounds | Star polygon | Compound | Star polygon | Compounds | ||
| Picture | .svg) {30/9}=3{10/3} | .svg) {30/10}=10{3} |  {30/11} | .svg) {30/12}=6{5/2} |  {30/13} | .svg) {30/14}=2{15/7} | .svg) {30/15}=15{2} |
| Interior angle | 72° | 60° | 48° | 36° | 24° | 12° | 0° |
There are alsoisogonal triacontagrams constructed as deeper truncations of the regularpentadecagon {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 thePetrie polygon for three 8-dimensional polytopes with E8 symmetry, shown inorthogonal projections in the E8Coxeter plane. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H4Coxeter plane.
| E8 | H4 | |||
|---|---|---|---|---|
 421 |  241 |  142 |  120-cell |  600-cell |
The regular triacontagram {30/7} is also the Petrie polygon for thegreat grand stellated 120-cell andgrand 600-cell.