User:Tomruen/Higher polygons

= Enneadecagon=

In geometry, an enneadecagon, enneakaidecagon, nonadecagon or 19-gon is a polygon with nineteen sides.

Regular form
A regular enneadecagon is represented by Schläfli symbol {19}.

The radius of the circumcircle of the regular enneadecagon with side length t is $$R=\frac{t}{2} \csc \frac {180}{19}$$ (angle in degrees). The area, where t is the edge length, is $$\frac{19}{4}t^2 \cot \frac{\pi}{19} \simeq 28.4652\,t^2.$$

Construction
As 19 is a Pierpont prime but not a Fermat prime, the regular enneadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector.
 * Approximated Enneadecagon Inscribed in a Circle.gif

Symmetry


The regular enneadecagon has Dih19 symmetry, order 38. Since 19 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z19, and Z1.

These 4 symmetries can be seen in 4 distinct symmetries on the enneadecagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r38 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g19 subgroup has no degrees of freedom but can seen as directed edges.

Related polygons
A enneadecagram is a 19-sided star polygon. There are eight regular forms given by Schläfli symbols: {19/2}, {19/3}, {19/4}, {19/5}, {19/6}, {19/7}, {19/8}, and {19/9}. Since 19 is prime, all enneadecagrams are regular stars and not compound figures.

Petrie polygons
The regular enneadecagon is the Petrie polygon for one higher-dimensional polytope, projected in a skew orthogonal projection:

Regular icosidigon
The regular icosidigon is represented by Schläfli symbol {22} and can also be constructed as a truncated hendecagon, t{11}.

The area of a regular icosidigon is: (with t = edge length)
 * $$A = \frac{11t^2}{2} \cot \frac{\pi}{22}.$$

Construction
As 22 = 2 × 11, the icosidigon can be constructed by truncating a regular hendecagon. However, the icosidigon is not constructible with a compass and straightedge, since 11 is not a Fermat prime. Consequently, the icosidigon cannot be constructed even with an angle trisector, because 11 is not a Pierpont prime. It can, however, be constructed with the neusis method.

Symmetry
The regular icosidigon has Dih22 symmetry, order 44. There are 3 subgroup dihedral symmetries: Dih11, Dih2, and Dih1, and 4 cyclic group symmetries: Z22, Z11, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the icosidigon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order. The full symmetry of the regular form is r44 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries n are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g22 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular icosidigons are d22, an isogonal icosidigon constructed by eleven mirrors which can alternate long and short edges, and p22, an isotoxal icosidigon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosidigon.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icosidigon, m=11, and it can be divided into 55: 5 sets of 11 rhombs. This decomposition is based on a Petrie polygon projection of a 11-cube.

Related polygons
An icosidigram is a 22-sided star polygon. There are 4 regular forms given by Schläfli symbols: {22/3}, {22/5}, {22/7}, and {22/9}. There are also 7 regular star figures using the same vertex arrangement: 2{11}, 11{2}.

There are also isogonal icosidigrams constructed as deeper truncations of the regular hendecagon {11} and hendecagrams {11/2}, {11/3}, {11/4} and {11/5}.

Regular icosihexagon
The regular icosihexagon is represented by Schläfli symbol {26} and can also be constructed as a truncated tridecagon, t{13}.

The area of a regular icosihexagon is: (with t = edge length)
 * $$A = 6.5t^2 \cot \frac{\pi}{26}.$$

Construction
As 26 = 2 × 13, the icosihexagon can be constructed by truncating a regular tridecagon. However, the icosihexagon is not constructible with a compass and straightedge, since 13 is not a Fermat prime. It can be constructed with an angle trisector, since 13 is a Pierpont prime.

Symmetry
The regular icosihexagon has Dih26 symmetry, order 52. There are 3 subgroup dihedral symmetries: Dih11, Dih2, and Dih1, and 4 cyclic group symmetries: Z26, Z13, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the icosihexagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order. The full symmetry of the regular form is r52 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries n are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g26 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular icosihexagons are d26, an isogonal icosihexagon constructed by thirteen mirrors which can alternate long and short edges, and p26, an isotoxal icosihexagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosihexagon.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icosihexagon, m=13, and it can be divided into 78: 6 sets of 13 rhombs. This decomposition is based on a Petrie polygon projection of a 13-cube.

Related polygons
An icosihexagram is a 26-sided star polygon. There are 5 regular forms given by Schläfli symbols: {26/3}, {26/5}, {26/7}, {26/9}, and {26/11}.

There are also isogonal icosihexagrams constructed as deeper truncations of the regular tridecagon {13} and tridecagrams {13/2}, {13/3}, {13/4}, {13/5} and {13/6}.

Regular icosioctagon
The regular icosioctagon is represented by Schläfli symbol {28} and can also be constructed as a truncated tetradecagon, t{14}, or a twice-truncated heptagon, tt{7}.

The area of a regular icosioctagon(28 sided polygon) is: (with t = edge length)
 * $$A = 7t^2 \cot \frac{\pi}{28}.$$

Construction
As 28 = 22 × 7, the icosioctagon is not constructible with a compass and straightedge, since 7 is not a Fermat prime. However, it can be constructed with an angle trisector, because 7 is a Pierpont prime.

Symmetry
The regular icosioctagon has Dih28 symmetry, order 56. There are 5 subgroup dihedral symmetries: (Dih14, Dih7), and (Dih4, Dih2, and Dih1), and 6 cyclic group symmetries: (Z28, Z14, Z7), and (Z4, Z2, Z1).

These 10 symmetries can be seen in 16 distinct symmetries on the icosioctagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order. The full symmetry of the regular form is r56 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g28 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular icosioctagons are d28, an isogonal icosioctagon constructed by ten mirrors which can alternate long and short edges, and p28, an isotoxal icosioctagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosioctagon.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m − 1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icosioctagon, m = 14, and it can be divided into 91: 7 squares and 6 sets of 14 rhombs. This decomposition is based on a Petrie polygon projection of a 14-cube.

Related polygons
An icosioctagram is a 28-sided star polygon. There are 5 regular forms given by Schläfli symbols: {28/3}, {28/5}, {28/9}, {28/11} and {28/13}.

There are also isogonal icosioctagrams constructed as deeper truncations of the regular tetradecagon {14} and tetradecagrams {28/3}, {28/5}, {28/9}, and {28/11}.

Regular triacontadigon
The regular triacontadigon can be constructed as a truncated hexadecagon, t{16}, a twice-truncated octagon, tt{8}, and a thrice-truncated square. A truncated triacontadigon, t{32}, is a hexacontatetragon, {64}.

One interior angle in a regular triacontadigon is 168$3/4$°, meaning that one exterior angle would be 11$1/4$°.

The area of a regular triacontadigon is (with )
 * $$\begin{align}

A = &8t^2 \cot \frac{\pi}{32}\\ = &8t^2 \left(1+ \sqrt{2} + \sqrt{4 +2 \sqrt{2} } +\sqrt{8 +4 \sqrt{2} +2 \sqrt{20 +14 \sqrt{2} } } \right) \end{align} $$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{32}$$

The circumradius of a regular triacontadigon is
 * $$\begin{align}

R = &\frac{1}{2}t \csc \frac{\pi}{32}\\ = &\frac{1}{2}t \left(\sqrt{16+8 \sqrt{2} +4 \sqrt{20 +14 \sqrt{2} } +2 \sqrt{168 +116 \sqrt{2} +2 \sqrt{13780 +9742 \sqrt{2} } } }\right) \end{align}$$

Construction
As 32 = 25 (a power of two), the regular triacontadigon is a constructible polygon. It can be constructed by an edge-bisection of a regular hexadecagon.

Symmetry
The regular triacontadigon has Dih32 dihedral symmetry, order 64, represented by 32 lines of reflection. Dih32 has 5 dihedral subgroups: Dih16, Dih8, Dih4, Dih2 and Dih1 and 6 more cyclic symmetries: Z32, Z16, Z8, Z4, Z2, and Z1, with Zn representing &pi;/n radian rotational symmetry.

On the regular triacontadigon, there are 17 distinct symmetries. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives r64 for the full reflective symmetry, Dih16, and a1 for no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular triacontadigons. Only the g32 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontadigon, m=16, and it can be divided into 120: 8 squares and 7 sets of 16 rhombs. This decomposition is based on a Petrie polygon projection of a 16-cube.

Triacontadigram
A triacontadigram is a 32-sided star polygon. There are seven regular forms given by Schläfli symbols {32/3}, {32/5}, {32/7}, {32/9}, {32/11}, {32/13}, and {32/15}, and eight compound star figures with the same vertex configuration.

Many isogonal triacontadigrams can also be constructed as deeper truncations of the regular hexadecagon {16} and hexadecagrams {16/3}, {16/5}, and {16/7}. These also create four quasitruncations: t{16/9} = {32/9}, t{16/11} = {32/11}, t{16/13} = {32/13}, and t{16/15} = {32/15}. Some of the isogonal triacontadigrams are depicted below as part of the aforementioned truncation sequences.

Regular triacontatetragon
A regular triacontatetragon is represented by Schläfli symbol {34} and can also be constructed as a truncated 17-gon, t{17}, which alternates two types of edges.

One interior angle in a regular triacontatetragon is (2880/17)°, meaning that one exterior angle would be (180/17)°.

The area of a regular triacontatetragon is (with )
 * $$A = \frac{17}{2}t^2 \cot \frac{\pi}{34}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{34}$$

The factor $$\cot \frac{\pi}{34}$$ is a root of the equation $$x^{16} - 136x^{14} + 2 380x^{12} - 12 376x^{10} + 24 310x^{8} - 19 448x^{6} + 6 188x^{4} - 680x^{2} + 17=0$$.

The circumradius of a regular triacontatetragon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{34}$$

As 34 = 2 × 17 and 17 is a Fermat prime, a regular triacontatetragon is constructible using a compass and straightedge. As a truncated 17-gon, it can be constructed by an edge-bisection of a regular 17-gon. This means that the values of $$\sin \frac{\pi}{34}$$ and $$\cos \frac{\pi}{34}$$ may be expressed in terms of nested radicals.

Symmetry
The regular triacontatetragon has Dih34 symmetry, order 68. There are 3 subgroup dihedral symmetries: Dih17, Dih2, and Dih1, and 4 cyclic group symmetries: Z34, Z17, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the icosidigon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order. The full symmetry of the regular form is labeled r68 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries n are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g34 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular triacontatetragons are d34, an isogonal triacontatetragon constructed by seventeen mirrors which can alternate long and short edges, and p34, an isotoxal triacontatetragon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular triacontatetragon.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontatetragon, m=17, it can be divided into 136: 8 sets of 17 rhombs. This decomposition is based on a Petrie polygon projection of a 17-cube.

Triacontatetragram
A triacontatetragram is a 34-sided star polygon. There are seven regular forms given by Schläfli symbols {34/3}, {34/5}, {34/7}, {34/9}, {34/11}, {34/13}, and {34/15}, and nine compound star figures with the same vertex configuration.

Many isogonal triacontatetragrams can also be constructed as deeper truncations of the regular heptadecagon {17} and heptadecagrams {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. These also create eight quasitruncations: t{17/9} = {34/9}, t{17/10} = {34/10}, t{17/11} = {34/11}, t{17/12} = {34/12}, t{17/13} = {34/13}, t{17/14} = {34/14}, t{17/15} = {34/15}, and t{17/16} = {34/16}. Some of the isogonal triacontatetragrams are depicted below, as a truncation sequence with endpoints t{17}={34} and t{17/16}={34/16}.

Regular tetracontagon
A regular tetracontagon is represented by Schlafli symbol {40} and can also be constructed as a truncated icosagon, t{20}, which alternates 2 types of edges. Furthermore, it can also be constructed as a twice-truncated decagon, tt{10}, or a thrice-truncated pentagon, ttt{5}.

One interior angle in a regular tetracontagon is 171°, meaning that one exterior angle would be 9°.

The area of a regular tetracontagon is (with )
 * $$\begin{align}A = 10t^2 \cot \frac{\pi}{40}=&10\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)^2+1}\right)t^2\\

=&10\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{\left(1+\sqrt{5}\right)^2+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}+\left(\sqrt{5+2\sqrt{5}}\right)^2+1}\right)t^2\\ =&10\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{\left(6+\binom{2}{1}\sqrt{5}\right)^{}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}+\left(5+2\sqrt{5}\right)^{}+1}\right)t^2\\ =&10\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{\left(11+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)+1}\right)t^2\\ =&10\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)t^2\end{align}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{40}$$

The factor $$\cot \frac{\pi}{40}$$ is a root of the octic equation $$x^{8} - 8x^{7} - 60x^{6} - 8x^{5} + 134x^{4} + 8x^{3} - 60x^{2} + 8x + 1$$.

The circumradius of a regular tetracontagon is
 * $$\begin{align}R = \frac{1}{2}t \csc \frac{\pi}{40}\end{align}$$

As 40 = 23 × 5, a regular tetracontagon is constructible using a compass and straightedge. As a truncated icosagon, it can be constructed by an edge-bisection of a regular icosagon. This means that the values of $$\sin \frac{\pi}{40}$$ and $$\cos \frac{\pi}{40}$$ may be expressed in radicals as follows:


 * $$\sin \frac{\pi}{40} = \frac{1}{4}(\sqrt{2}-1)\sqrt{\frac{1}{2}(2+\sqrt{2})(5+\sqrt{5})}-\frac{1}{8}\sqrt{2-\sqrt{2}}(1+\sqrt{2})(\sqrt{5}-1)$$
 * $$\cos \frac{\pi}{40} = \frac{1}{8}(\sqrt{2}-1)\sqrt{2+\sqrt{2}}(\sqrt{5}-1)+\frac{1}{4}(1+\sqrt{2})\sqrt{\frac{1}{2}(2-\sqrt{2})(5+\sqrt{5})}$$

Circumcircle is given

 * 1) Construct first the side length $\overline{JE_{1}}$ of a pentagon.
 * 2) Transfer this on the circumcircle, there arises the intersection E39.
 * 3) Connect the point E39 with the central point M, there arises the angle E39ME1 with 72°.
 * 4) Halve the angle E39ME1, there arise the intersection E40 and the angle E40ME1 with 9°.
 * 5) Connect the point E1 with the point E40, there arises the first side length a of the tetracontagon.
 * 6) Finally you transfer the segment E1E40 (side length a) repeatedly counterclockwise on the circumcircle until arises a regular tetracontagon.

The golden ratio
 * $$ \frac{\overline{JM}}{\overline{BJ}} = \frac{\overline{BM}}{\overline{JM}} = \frac{1 + \sqrt{5}}{2}= \varphi \approx 1.618 $$

Side length is given
[[File:01-Vierzigeck-Seitenlänge gegeben.svg|thumb|500px|Regular tetracontagon with given side length

(The construction is very similar to that of icosagon with given side length)]]


 * 1) Draw a segment $\overline{E_{40}E_{1}}$ whose length is the given side length a of the tetracontagon.
 * 2) Extend the segment $\overline{E_{40}E_{1}}$ by more than two times.
 * 3) Draw each a circular arc about the points E1 and E40, there arise the intersections A and B.
 * 4) Draw a vertical straight line from point B through point A.
 * 5) Draw a parallel line too the segment $\overline{AB}$ from the point E1 to the circular arc, there arises the intersection D.
 * 6) Draw a circle arc about the point C with the radius $\overline{CD}$ till to the extension of the side length, there arises the intersection F.
 * 7) Draw a circle arc about the point E40 with the radius $\overline{E_{40}F}$ till to the vertical straight line, there arises the intersection G and the angle E40GE1 with 36°.
 * 8) Draw a circle arc about the point G with radius $\overline{E_{40}G}$ till to the vertical straight line, there arises the intersection H and the angle E40HE1 with 18°.
 * 9) Draw a circle arc about the point H with radius $\overline{E_{40}H}$ till to the vertical straight line, there arises the central point M of the circumcircle and the angle E40ME1 with 9°.
 * 10) Draw around the central point M with radius $\overline{E_{40}M}$ the circumcircle of the tetracontagon.
 * 11) Finally transfer the segment $\overline{E_{40}E_{1}}$ (side length a) repeatedly counterclockwise on the circumcircle until to arises a regular tetracontagon.

The golden ratio
 * $$ \frac{\overline{E_{40}E_1}}{\overline{E_1F}} = \frac{\overline{E_{40}F}}{\overline{E_{40}E_1}} = \frac{1 + \sqrt{5}}{2}= \varphi \approx 1.618 $$

Symmetry
The regular tetracontagon has Dih40 dihedral symmetry, order 80, represented by 40 lines of reflection. Dih40 has 7 dihedral subgroups: (Dih20, Dih10, Dih5), and (Dih8, Dih4, Dih2, Dih1). It also has eight more cyclic symmetries as subgroups: (Z40, Z20, Z10, Z5), and (Z8, Z4, Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular tetracontagons. Only the g40 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m-cubes In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetracontagon, m=20, and it can be divided into 190: 10 squares and 9 sets of 20 rhombs. This decomposition is based on a Petrie polygon projection of a 20-cube.

Tetracontagram
A tetracontagram is a 40-sided star polygon. There are seven regular forms given by Schläfli symbols {40/3}, {40/7}, {40/9}, {40/11}, {40/13}, {40/17}, and {40/19}, and 12 compound star figures with the same vertex configuration.

Many isogonal tetracontagrams can also be constructed as deeper truncations of the regular icosagon {20} and icosagrams {20/3}, {20/7}, and {20/9}. These also create four quasitruncations: t{20/11}={40/11}, t{20/13}={40/13}, t{20/17}={40/17}, and t{20/19}={40/19}. Some of the isogonal tetracontagrams are depicted below, as a truncation sequence with endpoints t{20}={40} and t{20/19}={40/19}.

Regular tetracontadigon
The regular tetracontadigon can be constructed as a truncated icosihenagon, t{21}.

One interior angle in a regular tetracontadigon is 171$3/7$°, meaning that one exterior angle would be 8$4/7$°.

The area of a regular tetracontadigon is (with )
 * $$A = 10.5t^2 \cot \frac{\pi}{42}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{42}$$

The circumradius of a regular tetracontadigon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{42}$$

Since 42 = 2 × 3 × 7, a regular tetracontadigon is not constructible using a compass and straightedge, but is constructible if the use of an angle trisector is allowed.

Symmetry
The regular tetracontadigon has Dih42 dihedral symmetry, order 84, represented by 42 lines of reflection. Dih42 has 7 dihedral subgroups: Dih21, (Dih14, Dih7), (Dih6, Dih3), and (Dih2, Dih1) and 8 more cyclic symmetries: (Z42, Z21), (Z14, Z7), (Z6, Z3), and (Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

These 16 symmetries generate 20 unique symmetries on the regular tetracontadigon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives r84 for the full reflective symmetry, Dih42, and a1 for no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular tetracontadigons. Only the g42 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetracontadigon, m=21, it can be divided into 210: 10 sets of 21 rhombs. This decomposition is based on a Petrie polygon projection of a 21-cube.

Related polygons
An equilateral triangle, a regular heptagon, and a regular tetracontadigon can completely fill a plane vertex, one of 17 different combinations of regular polygons with this property. However, the entire plane cannot be tiled with regular polygons while including this vertex figure, though it can be used in a tiling with equilateral polygons and rhombi.

Tetracontadigram
A tetracontadigram is a 42-sided star polygon. There are five regular forms given by Schläfli symbols {42/5}, {42/11}, {42/13}, {42/17}, and {42/19}, as well as 15 compound star figures with the same vertex configuration.

Regular tetracontaoctagon
The regular tetracontaoctagon is represented by Schläfli symbol {48} and can also be constructed as a truncated icositetragon, t{24}, or a twice-truncated dodecagon, tt{12}, or a thrice-truncated hexagon, ttt{6}, or a fourfold-truncated triangle, tttt{3}.

One interior angle in a regular tetracontaoctagon is 172$1/2$°, meaning that one exterior angle would be 7$1/2$°.

The area of a regular tetracontaoctagon is: (with t = edge length)
 * $$\begin{align}

A &= 12t^2 \cot \frac{\pi}{48}\\ &= 12t^2 \left(2+\sqrt{3}+ \sqrt{ 8+4\sqrt{3} } + \sqrt{ 16+8\sqrt{3}+2 \sqrt{ 104+60\sqrt{3} } } \right) \\ &= 12t^2 \left(2+\sqrt{3}+( \sqrt{6} + \sqrt{2} )+ \sqrt{ 16+8\sqrt{3}+ 10\sqrt{2}+ 6\sqrt{6} } \right) \\ &= 12t^2 \left(2+\sqrt{3}+( \sqrt{6} + \sqrt{2} )+ 2\sqrt{ 4+2\sqrt{3}+ \sqrt{ 26+15\sqrt{3} } } \right). \end{align}$$

The tetracontaoctagon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), icositetragon (24-gon), and enneacontahexagon (96-gon).

Construction
Since 48 = 24 × 3, a regular tetracontaoctagon is constructible using a compass and straightedge. As a truncated icositetragon, it can be constructed by an edge-bisection of a regular icositetragon.

Symmetry
The regular tetracontaoctagon has Dih48 symmetry, order 96. There are nine subgroup dihedral symmetries: (Dih24, Dih12, Dih6, Dih3), and (Dih16, Dih8, Dih4, Dih2 Dih1), and 10 cyclic group symmetries: (Z48, Z24, Z12, Z6, Z3), and (Z16, Z8, Z4, Z2, Z1).

These 20 symmetries can be seen in 28 distinct symmetries on the tetracontaoctagon. John Conway labels these by a letter and group order. The full symmetry of the regular form is r96 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g48 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetracontaoctagon, m=24, and it can be divided into 276: 12 squares and 11 sets of 24 rhombs. This decomposition is based on a Petrie polygon projection of a 24-cube.

Tetracontaoctagram
A tetracontaoctagram is a 48-sided star polygon. There are seven regular forms given by Schläfli symbols {48/5}, {48/7}, {48/11}, {48/13}, {48/17}, {48/19}, and {48/23}, as well as 16 compound star figures with the same vertex configuration.

Regular pentacontagon properties
One interior angle in a regular pentacontagon is 172$4/5$°, meaning that one exterior angle would be 7$1/5$°.

The area of a regular pentacontagon is (with )
 * $$A = \frac{25}{2}t^2 \cot \frac{\pi}{50}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{50}$$

The circumradius of a regular pentacontagon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{50}$$

Since 50 = 2 × 52, a regular pentacontagon is not constructible using a compass and straightedge, and is not constructible even if the use of an angle trisector is allowed. However, it is constructible using an auxiliary curve (such as the quadratrix of Hippias or an Archimedean spiral), as such curves can be used to divide angles into any number of equal parts. For example, one can construct a 36° angle using compass and straightedge and proceed to quintisect it (divide it into five equal parts) using an Archimedean spiral, giving the 7.2° angle required to construct a pentacontagon.

It is not known if the pentacontagon is neusis-constructible.

Symmetry
The regular pentacontagon has Dih50 dihedral symmetry, order 100, represented by 50 lines of reflection. Dih50 has 5 dihedral subgroups: Dih25, (Dih10, Dih5), and (Dih2, Dih1). It also has 6 more cyclic symmetries as subgroups: (Z50, Z25), (Z10, Z5), and (Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedom in defining irregular pentacontagons. Only the g50 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular pentacontagon, m=25, it can be divided into 300: 12 sets of 25 rhombs. This decomposition is based on a Petrie polygon projection of a 25-cube.

Pentacontagram
A pentacontagram is a 50-sided star polygon. There are 9 regular forms given by Schläfli symbols {50/3}, {50/7}, {50/9}, {50/11}, {50/13}, {50/17}, {50/19}, {50/21}, and {50/23}, as well as 16 compound star figures with the same vertex configuration.

Regular hexacontagon properties
A regular hexacontagon is represented by Schläfli symbol {60} and also can be constructed as a truncated triacontagon, t{30}, or a twice-truncated pentadecagon, tt{15}. A truncated hexacontagon, t{60}, is a 120-gon, {120}.

One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°.

The area of a regular hexacontagon is (with )
 * $$A = 15t^2 \cot \frac{\pi}{60}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{60}$$

The circumradius of a regular hexacontagon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{60}$$

This means that the trigonometric functions of π/60 can be expressed in radicals.

Constructible
Since 60 = 22 × 3 × 5, a regular hexacontagon is constructible using a compass and straightedge. As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon.

Symmetry
The regular hexacontagon has Dih60 dihedral symmetry, order 120, represented by 60 lines of reflection. Dih60 has 11 dihedral subgroups: (Dih30, Dih15), (Dih20, Dih10, Dih5), (Dih12, Dih6, Dih3), and (Dih4, Dih2, Dih1). And 12 more cyclic symmetries: (Z60, Z30, Z15), (Z20, Z10, Z5), (Z12, Z6, Z3), and (Z4, Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

These 24 symmetries are related to 32 distinct symmetries on the hexacontagon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedom in defining irregular hexacontagons. Only the g60 symmetry has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexacontagon, m=30, and it can be divided into 435: 15 squares and 14 sets of 30 rhombs. This decomposition is based on a Petrie polygon projection of a 30-cube.

Hexacontagram
A hexacontagram is a 60-sided star polygon. There are 7 regular forms given by Schläfli symbols {60/7}, {60/11}, {60/13}, {60/17}, {60/19}, {60/23}, and {60/29}, as well as 22 compound star figures with the same vertex configuration.

Regular hexacontatetragon
The regular hexacontatetragon can be constructed as a truncated triacontadigon, t{32}, a twice-truncated hexadecagon, tt{16}, a thrice-truncated octagon, ttt{8}, a fourfold-truncated square, tttt{4}, and a fivefold-truncated digon, ttttt{2}.

One interior angle in a regular hexacontatetragon is 174$50/3$°, meaning that one exterior angle would be 5$50/7$°.

The area of a regular hexacontatetragon is (with )
 * $$A = 16t^2 \cot \frac{\pi}{64}$$

and its inradius is
 * $$r = \frac{t}{2} \cot \frac{\pi}{64}$$

The circumradius of a regular hexacontatetragon is
 * $$R = \frac{t}{2} \csc \frac{\pi}{64}$$

Construction
Since 64 = 26 (a power of two), a regular hexacontatetragon is constructible using a compass and straightedge. As a truncated triacontadigon, it can be constructed by an edge-bisection of a regular triacontadigon.

Symmetry
The regular hexacontatetragon has Dih64 dihedral symmetry, order 128, represented by 64 lines of reflection. Dih64 has 6 dihedral subgroups: Dih32, Dih16, Dih8, Dih4, Dih2 and Dih1 and 7 more cyclic symmetries: Z64, Z32, Z16, Z8, Z4, Z2, and Z1, with Zn representing &pi;/n radian rotational symmetry.

These 13 symmetries generate 20 unique symmetries on the regular hexacontatetragon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives r128 for the full reflective symmetry, Dih64, and a1 for no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular hexacontatetragons. Only the g64 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m−1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexacontatetragon, m=32, and it can be divided into 496: 16 squares and 15 sets of 32 rhombs. This decomposition is based on a Petrie polygon projection of a 32-cube.

Hexacontatetragram
A hexacontatetragram is a 64-sided star polygon. There are 15 regular forms given by Schläfli symbols {64/3}, {64/5}, {64/7}, {64/9}, {64/11}, {64/13}, {64/15}, {64/17}, {64/19}, {64/21}, {64/23}, {64/25}, {64/27}, {64/29}, {64/31}, as well as 16 compound star figures with the same vertex configuration.

Regular heptacontagon properties
One interior angle in a regular heptacontagon is 174$50/9$°, meaning that one exterior angle would be 5$50/11$°.

The area of a regular heptacontagon is (with )
 * $$A = \frac{35}{2}t^2 \cot \frac{\pi}{70}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{70}$$

The circumradius of a regular heptacontagon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{70}$$

Since 70 = 2 × 5 × 7, a regular heptacontagon is not constructible using a compass and straightedge, but is constructible if the use of an angle trisector is allowed.

Symmetry
The regular heptacontagon has Dih70 dihedral symmetry, order 140, represented by 70 lines of reflection. Dih70 has 7 dihedral subgroups: Dih35, (Dih14, Dih7), (Dih10, Dih5), and (Dih2, Dih1). It also has 8 more cyclic symmetries as subgroups: (Z70, Z35), (Z14, Z7), (Z10, Z5), and (Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular heptacontagons. Only the g70 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular heptacontagon, m=35, it can be divided into 595: 17 sets of 35 rhombs. This decomposition is based on a Petrie polygon projection of a 35-cube.

Heptacontagram
A heptacontagram is a 70-sided star polygon. There are 11 regular forms given by Schläfli symbols {70/3}, {70/9}, {70/11}, {70/13}, {70/17}, {70/19}, {70/23}, {70/27}, {70/29}, {70/31}, and {70/33}, as well as 23 regular star figures with the same vertex configuration.

Regular octacontagon
A regular octacontagon is represented by Schläfli symbol {80} and can also be constructed as a truncated tetracontagon, t{40}, or a twice-truncated icosagon, tt{20}, or a thrice-truncated decagon, ttt{10}, or a four-fold-truncated pentagon, tttt{5}.

One interior angle in a regular octacontagon is 175$50/13$°, meaning that one exterior angle would be 4$50/17$°.

The area of a regular octacontagon is (with )
 * $$\begin{align}A = 20t^2 \cot \frac{\pi}{80}=\cot\frac{\pi}{40}+\sqrt{\cot^2\frac{\pi}{40}+1}=& 20\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}+\sqrt{\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)^2+1}\right)t^2\\

=& 20\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}+\sqrt{\left(\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)+\left(\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)\right)^2+1}\right)t^2\\ =&20\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}+\sqrt{\left(\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)^2+\binom{2}{1}\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)\left(\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)+\left(\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)^2\right)+1}\right)t^2\\ =&20\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}+\sqrt{\left(\left(11+4\sqrt5+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)+\binom{2}{1}\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)\left(\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)+\left(12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)\right)+1}\right)t^2\\ =&20\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}+\sqrt{\left(23+8\sqrt5+2\cdot\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}+\binom{2}{1}\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)\left(\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)\right)+1}\right)t^2\\ =&20\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}+\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}+\sqrt{24+8\sqrt5+2\cdot\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}+\binom{2}{1}\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)\left(\sqrt{12+4\sqrt{5}+\binom{2}{1}\left(1+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}}\right)}\right)t^2\end{align}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{80}$$

The circumradius of a regular octacontagon is
 * $$\begin{align}R = \frac{1}{2}t \csc \frac{\pi}{80}=\frac{\sqrt{\left(\cot\tfrac{\pi}{80}\right)^2+1}}{2}t\end{align}$$

Construction
Since 80 = 24 × 5, a regular octacontagon is constructible using a compass and straightedge. As a truncated tetracontagon, it can be constructed by an edge-bisection of a regular tetracontagon. This means that the trigonometric functions of π/80 can be expressed in radicals:


 * $$\sin\frac{\pi}{80}=\sin 2.25^\circ=\frac{1}{8}(1+\sqrt{5})\left(-\sqrt{2-\sqrt{2+\sqrt{2}}}-\sqrt{(2+\sqrt{2})\left(2-\sqrt{2+\sqrt{2}}\right)}\right)$$
 * $$+\frac{1}{4}\sqrt{\frac{1}{2}(5-\sqrt{5})}\left(\sqrt{(2+\sqrt{2})\left(2+\sqrt{2+\sqrt{2}}\right)}-\sqrt{2+\sqrt{2+\sqrt{2}}}\right)$$
 * $$\cos\frac{\pi}{80}=\cos 2.25^\circ=\sqrt{\frac{1}{2}+\frac{1}{4}\sqrt{\frac{1}{2}\left(4+\sqrt{2\left(4+\sqrt{2(5+\sqrt{5})}\right)}\right)}}$$

Symmetry
The regular octacontagon has Dih80 dihedral symmetry, order 80, represented by 80 lines of reflection. Dih40 has 9 dihedral subgroups: (Dih40, Dih20, Dih10, Dih5), and (Dih16, Dih8, Dih4, and Dih2, Dih1). It also has 10 more cyclic symmetries as subgroups: (Z80, Z40, Z20, Z10, Z5), and (Z16, Z8, Z4, Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. r160 represents full symmetry and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry.

These lower symmetries allows degrees of freedoms in defining irregular octacontagons. Only the g80 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octacontagon, m=40, and it can be divided into 780: 20 squares and 19 sets of 40 rhombs. This decomposition is based on a Petrie polygon projection of a 40-cube.

Octacontagram
An octacontagram is an 80-sided star polygon. There are 15 regular forms given by Schläfli symbols {80/3}, {80/7}, {80/9}, {80/11}, {80/13}, {80/17}, {80/19}, {80/21}, {80/23}, {80/27}, {80/29}, {80/31}, {80/33}, {80/37}, and {80/39}, as well as 24 regular star figures with the same vertex configuration.

Regular enneacontagon properties
One interior angle in a regular enneacontagon is 176°, meaning that one exterior angle would be 4°.

The area of a regular enneacontagon is (with )
 * $$A = \frac{45}{2}t^2 \cot \frac{\pi}{90}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{90}$$

The circumradius of a regular enneacontagon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{90}$$

Since 90 = 2 × 32 × 5, a regular enneacontagon is not constructible using a compass and straightedge, but is constructible if the use of an angle trisector is allowed.

Symmetry
The regular enneacontagon has Dih90 dihedral symmetry, order 180, represented by 90 lines of reflection. Dih90 has 11 dihedral subgroups: Dih45, (Dih30, Dih15), (Dih18, Dih9), (Dih10, Dih5), (Dih6, Dih3), and (Dih2, Dih1). And 12 more cyclic symmetries: (Z90, Z45), (Z30, Z15), (Z18, Z9), (Z10, Z5), (Z6, Z3), and (Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

These 24 symmetries are related to 30 distinct symmetries on the enneacontagon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedom in defining irregular enneacontagons. Only the g90 symmetry has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular enneacontagon, m=45, it can be divided into 990: 22 sets of 45 rhombs. This decomposition is based on a Petrie polygon projection of a 45-cube.

Enneacontagram
An enneacontagram is a 90-sided star polygon. There are 11 regular forms given by Schläfli symbols {90/7}, {90/11}, {90/13}, {90/17}, {90/19}, {90/23}, {90/29}, {90/31}, {90/37}, {90/41}, and {90/43}, as well as 33 regular star figures with the same vertex configuration.

Regular enneacontahexagon
The regular enneacontahexagon is represented by Schläfli symbol {96} and can also be constructed as a truncated tetracontaoctagon, t{48}, or a twice-truncated icositetragon, tt{24}, or a thrice-truncated dodecagon, ttt{12}, or a fourfold-truncated hexagon, tttt{6}, or a fivefold-truncated triangle, ttttt{3}.

One interior angle in a regular enneacontahexagon is 176$50/19$°, meaning that one exterior angle would be 3$50/21$°.

The area of a regular enneacontahexagon is: (with t = edge length)
 * $$A = 24t^2 \cot \frac{\pi}{96}$$

The enneacontahexagon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), icositetragon (24-gon), and tetracontaoctagon (48-gon).

Construction
Since 96 = 25 × 3, a regular enneacontahexagon is constructible using a compass and straightedge. As a truncated tetracontaoctagon, it can be constructed by an edge-bisection of a regular tetracontaoctagon.

Symmetry
The regular enneacontahexagon has Dih96 symmetry, order 192. There are 11 subgroup dihedral symmetries: (Dih48, Dih24, Dih12, Dih6, Dih3), (Dih32, Dih16, Dih8, Dih4, Dih2 and Dih1), and 12 cyclic group symmetries: (Z96, Z48, Z24, Z12, Z6, Z3), (Z32, Z16, Z8, Z4, Z2, and Z1).

These 24 symmetries can be seen in 34 distinct symmetries on the enneacontahexagon. John Conway labels these by a letter and group order. The full symmetry of the regular form is r192 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g96 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular enneacontahexagon, m=48, and it can be divided into 1128: 24 squares and 23 sets of 48 rhombs. This decomposition is based on a Petrie polygon projection of a 48-cube.

Enneacontahexagram
An enneacontahexagram is a 96-sided star polygon. There are 15 regular forms given by Schläfli symbols {96/5}, {96/7}, {96/11}, {96/13}, {96/17}, {96/19}, {96/23}, {96/25}, {96/29}, {96/31}, {96/35}, {96/37}, {96/41}, {96/43}, and {96/47}, as well as 32 compound star figures with the same vertex configuration.

Regular hectogon
A regular hectogon is represented by Schläfli symbol {100} and can be constructed as a truncated pentacontagon, t{50}, or a twice-truncated icosipentagon, tt{25}.

One interior angle in a regular hectogon is 176$50/23$°, meaning that one exterior angle would be 3$3/8$°.

The area of a regular hectogon is (with )
 * $$A = 25t^2 \cot \frac{\pi}{100}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{100}$$

The circumradius of a regular hectogon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{100}$$

Because 100 = 22 &times; 52, the number of sides contains a repeated Fermat prime (the number 5). Thus the regular hectogon is not a constructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.

It is not known if the regular hectogon is neusis constructible. Its neusis constructibility is equivalent to that of the 25-gon, which is an open problem.

However, a hectogon is constructible using an auxiliary curve such as an Archimedean spiral. A 72° angle is constructible with compass and straightedge, so a possible approach to constructing one side of a hectogon is to construct a 72° angle using compass and straightedge, use an Archimedean spiral to construct a 14.4° angle, and bisect one of the 14.4° angles twice.

Symmetry
The regular hectogon has Dih100 dihedral symmetry, order 200, represented by 100 lines of reflection. Dih100 has 8 dihedral subgroups: (Dih50, Dih25), (Dih20, Dih10, Dih5), (Dih4, Dih2, and Dih1). It also has 9 more cyclic symmetries as subgroups: (Z100, Z50, Z25), (Z20, Z10, Z5), and (Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. r200 represents full symmetry and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry.

These lower symmetries allows degrees of freedom in defining irregular hectogons. Only the g100 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hectogon, m=50, it can be divided into 1225: 25 squares and 24 sets of 50 rhombs. This decomposition is based on a Petrie polygon projection of a 50-cube.

Hectogram
A hectogram is a 100-sided star polygon. There are 19 regular forms given by Schläfli symbols {100/3}, {100/7}, {100/9}, {100/11}, {100/13}, {100/17}, {100/19}, {100/21}, {100/23}, {100/27}, {100/29}, {100/31}, {100/33}, {100/37}, {100/39}, {100/41}, {100/43}, {100/47}, and {100/49}, as well as 30 regular star figures with the same vertex configuration.

Regular 120-gon properties
A regular 120-gon is represented by Schläfli symbol {120} and also can be constructed as a truncated hexacontagon, t{60}, or a twice-truncated triacontagon, tt{30}, or a thrice-truncated pentadecagon, ttt{15}.

One interior angle in a regular 120-gon is 177°, meaning that one exterior angle would be 3°.

The area of a regular 120-gon is (with )
 * $$A = 30t^2 \cot \frac{\pi}{120}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{120}$$

The circumradius of a regular 120-gon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{120}$$

This means that the trigonometric functions of π/120 can be expressed in radicals.

Constructible
Since 120 = 23 × 3 × 5, a regular 120-gon is constructible using a compass and straightedge. As a truncated hexacontagon, it can be constructed by an edge-bisection of a regular hexacontagon.

Symmetry
The regular 120-gon has Dih120 dihedral symmetry, order 240, represented by 120 lines of reflection. Dih120 has 15 dihedral subgroups: (Dih60, Dih30, Dih15), (Dih40, Dih20, Dih10, Dih5), (Dih24, Dih12, Dih6, Dih3), and (Dih8, Dih4, Dih2, Dih1). And 16 more cyclic symmetries: (Z120, Z60, Z30, Z15), (Z40, Z20, Z10, Z5), (Z24, Z12, Z6, Z3), and (Z8, Z4, Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

These 32 symmetries are related to 44 distinct symmetries on the 120-gon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allow degrees of freedom in defining irregular 120-gons. Only the g120 symmetry has no degrees of freedom but can be seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular 120-gon, m=60, and it can be divided into 1770: 30 squares and 29 sets of 60 rhombs. This decomposition is based on a Petrie polygon projection of a 60-cube.

120-gram
A 120-gram is a 120-sided star polygon. There are 15 regular forms given by Schläfli symbols {120/7}, {120/11}, {120/13}, {120/17}, {120/19}, {120/23}, {120/29}, {120/31}, {120/37}, {120/41}, {120/43}, {120/47}, {120/49}, {120/53}, and {120/59}, as well as 44 compound star figures with the same vertex configuration.

Regular 360-gon
A regular 360-gon is represented by Schläfli symbol {360} and also can be constructed as a truncated 180-gon, t{180}, or a twice-truncated enneacontagon, tt{90}, or a thrice-truncated tetracontapentagon, ttt{45}.

One interior angle in a regular 360-gon is 179°, meaning that one exterior angle would be 1°.

The area of a regular 360-gon is (with )
 * $$A = 90t^2 \cot \frac{\pi}{360}$$

and its inradius is
 * $$r = \frac{1}{2}t \cot \frac{\pi}{360}$$

The circumradius of a regular 360-gon is
 * $$R = \frac{1}{2}t \csc \frac{\pi}{360}$$

Since 360 = 23 × 32 × 5, a regular 360-gon is not constructible using a compass and straightedge, but is constructible if the use of an angle trisector is allowed.

Symmetry
The regular 360-gon has Dih360 dihedral symmetry, order 720, represented by 360 lines of reflection. Dih360 has 23 dihedral subgroups: (Dih180, Dih90, Dih45), (Dih120, Dih60, Dih30, Dih15), (Dih72, Dih36, Dih18, Dih9), (Dih40, Dih20, Dih10, Dih5), (Dih24, Dih12, Dih6, Dih3), and (Dih8, Dih4, Dih2, Dih1). And 24 more cyclic symmetries: (Z360, Z180, Z90, Z45), (Z120, Z60, Z30, Z15), (Z72, Z36, Z18, Z9), (Z40, Z20, Z10, Z5), (Z24, Z12, Z6, Z3), and (Z8, Z4, Z2, Z1), with Zn representing &pi;/n radian rotational symmetry.

These 48 symmetries are related to 66 distinct symmetries on the 360-gon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. Full symmetry is r720 and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry.

These lower symmetries allows degrees of freedom in defining irregular 360-gons. Only the g360 symmetry has no degrees of freedom but can seen as directed edges.

Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular 360-gon, m=180, and it can be divided into 16110: 90 squares and 89 sets of 180 rhombs. This decomposition is based on a Petrie polygon projection of a 180-cube.

360-gram
A 360-gram is a 360-sided star polygon. There are 47 regular forms given by Schläfli symbols {360/7}, {360/11}, {360/13}, {360/17}, {360/19}, {360/23}, {360/29}, {360/31}, {360/37}, {360/41}, {360/43}, {360/47}, {360/49}, {360/53}, {360/59}, {360/61}, {360/67}, {360/71}, {360/73}, {360/77}, {360/79}, {360/83}, {360/89}, {360/91}, {360/97}, {360/101}, {360/103}, {360/107}, {360/109}, {360/113}, {360/119}, {360/121}, {360/127}, {360/131}, {360/133}, {360/137}, {360/139}, {360/143}, {360/149}, {360/151}, {360/157}, {360/161}, {360/163}, {360/167}, {360/169}, {360/173}, and {360/179}, as well as 132 compound star figures with the same vertex configuration. Many of the more intricate 360-grams show moiré patterns. The regular convex and star polygons whose interior angles are some integer number of degrees are precisely those whose numbers of sides are integer divisors of 360 that are not unity, i.e. {2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360}.