Triacontagon

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}.

Regular triacontagon properties

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)

\( A = \frac{15}{2} t^2 \cot \frac{\pi}{30} = \frac{15}{2} t^2 (\sqrt{23 + 10 \sqrt{5} + 2 \sqrt{3(85 + 38 \sqrt{5})}} = \frac{15}{4} t^2 (\sqrt{15} + 3\sqrt{3} + \sqrt{2}\sqrt{25+11\sqrt{5}}) \)

The inradius of a regular triacontagon is

\( r = \frac{1}{2} t \cot \frac{\pi}{30} = \frac{1}{4} t(\sqrt{15} + 3\sqrt{3} + \sqrt{2}\sqrt{25+11\sqrt{5}}) \)

The circumradius of a regular triacontagon is

\( R = \frac{1}{2} t \csc \frac{\pi}{30} = \frac{1}{2} t(2 + \sqrt{5} + \sqrt{15+6\sqrt{5}}) \)

Construction

A regular triacontagon is constructible using a compass and straightedge.[1]
Triacontagram

A triacontagram is a 30-sided star polygon. 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.

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}.[2]

Petrie polygons

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.

The regular triacontagram {30/7} is also the Petrie polygon for the great grand stellated 120-cell and grand 600-cell.
References

Constructible Polygon

The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

Weisstein, Eric W., "Triacontagon", MathWorld.
Naming Polygons and Polyhedra
triacontagon

Mathematics Encyclopedia