Triacontagon

Regular triacontagon
A regular triacontagon
Type	Regular polygon
Edges and vertices	30
Schläfli symbol	{30}, t{15}
Coxeter diagram
Symmetry group	Dihedral (D30), order 2×30
Internal angle (degrees)	168°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

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°, 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)

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.

The regular 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 more cyclic symmetries as subgroups: (Z₃₀, Z₁₅), (Z₁₀, Z₅), (Z₆, Z₃), and (Z₂, Z₁), with Z_n representing π/n radian rotational symmetry.