Triacontagon
polygon with 30 sides
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
Regular 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}. A truncated triacontagon, t{30}, is a hexacontagon, {60}.
One interior angle in a regular triacontagon is 168 degrees, 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
=
15
2
t
2
cot
π
30
=
15
4
t
2
(
15
+
3
3
+
2
25
+
11
5
)
{\displaystyle A={\frac {15}{2}}t^{2}\cot {\frac {\pi }{30}}={\frac {15}{4}}t^{2}\left({\sqrt {15}}+3{\sqrt {3}}+{\sqrt {2}}{\sqrt {25+11{\sqrt {5}}}}\right)}
The inradius of a regular triacontagon is
r
=
1
2
t
cot
π
30
=
1
4
t
(
15
+
3
3
+
2
25
+
11
5
)
{\displaystyle r={\frac {1}{2}}t\cot {\frac {\pi }{30}}={\frac {1}{4}}t\left({\sqrt {15}}+3{\sqrt {3}}+{\sqrt {2}}{\sqrt {25+11{\sqrt {5}}}}\right)}
The circumradius of a regular triacontagon is
R
=
1
2
t
csc
π
30
=
1
2
t
(
2
+
5
+
15
+
6
5
)
{\displaystyle R={\frac {1}{2}}t\csc {\frac {\pi }{30}}={\frac {1}{2}}t\left(2+{\sqrt {5}}+{\sqrt {15+6{\sqrt {5}}}}\right)}
Construction
As 30 = 2 × 3 × 5 , a regular triacontagon is constructible using a compass and straightedge.
Symmetry
The regular triacontagon has Dih30 dihedral 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 Zn representing π/n radian rotational symmetry.
The public source identifies “Triacontagon” as polygon with 30 sides. This brief keeps that definition visible, then builds a research path around Triacontagon, polygon and sides.
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