Triacontagon

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

In geometry, an triacontagon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

One interior angle in a regular triacontagon is 168° meaning that one exterior angle would be 12°.

The regular triacontagon is a constructible polygon, by an edge-bisection of a regular pentadecagon, and can be seen as a truncated pentadecagon.

Area

The area of a regular triacontangon is (with t = edge length)

$A = \frac{15}{2} t^2 \cot \frac{\pi}{30}$

Petrie polygons

The regular triacontagon is the Petrie polygon for a number of higher dimensional polytopes with E₈ symmetry, shown in orthogonal projections in the E₈ Coxeter plane:

(421)

t1(421)

t2(421)

(241)

t1(241)

It is also the Petrie polygon for some higher dimensional polytopes with H₄ symmetry, shown in orthogonal projections in the H₄ Coxeter plane:

120-cell

Rectified 120-cell

Rectified 600-cell

600-cell

References

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