# Enneagon

Enneagon Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymEn
Coxeter diagramx9o (   )
Schläfli symbol{9}
Elements
Edges9
Vertices9
Vertex figureDyad, length $2\cos(\pi/9)$ Measures (edge length 1)
Circumradius$\frac{1}{2\sin\frac\pi9} \approx 1.46190$ Inradius$\frac{1}{2\tan\frac\pi9} \approx 1.37374$ Area$\frac{9}{4\tan\frac\pi9} \approx 6.18182$ 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
ConvexYes
Net count1
NatureTame

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

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

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".

## Vertex coordinates

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

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

## 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.