Triacontagon

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Regular triacontagon
Regular polygon 30 vertex animation.svg
A regular triacontagon
Type Regular polygon
Edges and vertices 30
Schläfli symbol {30}
Coxeter diagram CDel node 1.pngCDel 3x.pngCDel 0x.pngCDel node.png
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 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 regular triacontagon is a constructible polygon, by an edge-bisection of a regular pentadecagon, and can be seen as a truncated pentadecagon.

Area[edit]

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

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

Petrie polygons[edit]

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:

4 21 t0 E8.svg
(421)
4 21 t1 E8.svg
t1(421)
4 21 t2 E8.svg
t2(421)
2 41 t0 E8.svg
(241)
2 41 t1 E8.svg
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 graph H4.svg
120-cell
120-cell t1 H4.svg
Rectified 120-cell
600-cell t1 H4.svg
Rectified 600-cell
600-cell graph H4.svg
600-cell

References[edit]