# Triacontagon

Triacontagon
Rank2
TypeRegular
Notation
Coxeter diagramx30o ()
Schläfli symbol{30}
Elements
Edges30
Vertices30
Vertex figureDyad, length ${\displaystyle {\frac {\sqrt {7+{\sqrt {5}}+{\sqrt {30+6{\sqrt {5}}}}}}{2}}}$
Measures (edge length 1)
Circumradius${\displaystyle {\frac {2+{\sqrt {5}}+{\sqrt {15+6{\sqrt {5}}}}}{2}}\approx 4.78339}$
Inradius${\displaystyle {\frac {\sqrt {23+10{\sqrt {5}}+2{\sqrt {255+114{\sqrt {5}}}}}}{2}}\approx 4.75718}$
Area${\displaystyle {\frac {15{\sqrt {23+10{\sqrt {5}}+2{\sqrt {255+114{\sqrt {5}}}}}}}{2}}\approx 71.35773}$
Angle${\displaystyle {\frac {14\pi }{15}}=168^{\circ }}$
Central density1
Number of external pieces30
Level of complexity1
Related polytopes
DualTriacontagon
Conjugate3 total
Abstract & topological properties
Flag count60
Euler characteristic0
OrientableYes
Properties
SymmetryI2(30), order 60
Flag orbits1
ConvexYes
NatureTame

The triacontagon is a polygon with 30 sides. A regular triacontagon has equal sides and equal angles.

This polygon is notable for the property that there exist sets of 7 diagonals, all concurrent in a point other than the center. No other regular polygon has more concurrent diagonals, and any other polygon with this property must have a multiple of 30 sides.[1]

Since 30 = 2 × 3 × 5 is the product of different Fermat primes, a regular triacontagon is constructible with straightedge and compass.

The interior angles of an equilateral triangle and a regular pentagon add up to that of the 30-gon: ${\displaystyle \theta (3)+\theta (5)=\theta (30)}$ where ${\displaystyle \theta (n)={\frac {n-2}{n}}\pi }$. Only two other finite convex regular polygons have internal angles expressible as such sums: the hexagon ${\displaystyle \theta (3)+\theta (3)=\theta (6)}$ and the dodecagon ${\displaystyle \theta (3)+\theta (4)=\theta (12)}$. Stewart used this construct in a family of quasiconvex Stewart toroids; as an example, a virtual 30-gonal prism can be lined on the inside with alternating triangular and pentagonal prisms, and the coplanar faces can be resolved by excavating pyramids.