# Enneagon

The enneagon sometimes referred to as a nonagon, is a polygon with 9 sides. A regular enneagon has equal sides and equal angles.

Enneagon
Rank2
TypeRegular
Notation
Bowers style acronymEn
Coxeter diagramx9o ()
Schläfli symbol{9}
Elements
Edges9
Vertices9
Vertex figureDyad, length ${\displaystyle 2\cos(\pi /9)}$
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1}{2\sin {\frac {\pi }{9}}}}\approx 1.46190}$
Inradius${\displaystyle {\frac {1}{2\tan {\frac {\pi }{9}}}}\approx 1.37374}$
Area${\displaystyle {\frac {9}{4\tan {\frac {\pi }{9}}}}\approx 6.18182}$
Angle140°
Central density1
Number of external pieces9
Level of complexity1
Related polytopes
ArmyEn
DualEnneagon
ConjugatesEnneagram, great enneagram
Abstract & topological properties
Flag count18
Euler characteristic0
OrientableYes
Properties
SymmetryI2(9), order 18
Flag orbits1
ConvexYes
Net count1
NatureTame

Like regular heptagons, regular enneagons are rarely found in higher polytopes that are objects of study, as they do not occur any non-prismatic uniform polyhedra or Johnson solids. A notable exception is the pairwise augmented cupolas, which are acrohedra. Enneagons also appear in some near-miss Johnson solids, such as the sesquitruncated octahedron.

## Naming

The name enneagon is derived from the Ancient Greek ἐννέα (9) and γωνία (angle), referring to the number of vertices.

Other names include:

• En, Bowers style acronym, short for "enneagon".

The combining prefix in BSAs is e-, as in edip.

## Vertex coordinates

Coordinates for an enneagon of edge length ${\displaystyle 2\sin(\pi /9)}$ , centered at the origin, are:

• ${\displaystyle (1,0)}$ ,
• ${\displaystyle (\cos(2\pi /9),\pm \sin(2\pi /9))}$ ,
• ${\displaystyle (\cos(4\pi /9),\pm \sin(4\pi /9))}$ ,
• ${\displaystyle (-1/2,\pm {\sqrt {3}}/2)}$ ,
• ${\displaystyle (\cos(8\pi /9),\pm \sin(8\pi /9))}$ .

## Variations

Besides the regular enneagon, other enneagons with triangular, mirror, or no symmetry exist. A few higher polytopes, such as certain swirlchora, have trigon-symmetric enneagons as faces.