Nonagon

Regular nonagon
A regular nonagon
Edges and vertices	9
Schläfli symbol	{9}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D9)
Internal angle (degrees)	140°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a nonagon  /ˈnɒnəɡɒn/ (or enneagon /ˈɛniːəɡɒn/) is a nine-sided polygon.

The name "nonagon" is a prefix hybrid formation, from Latin (nonus, "ninth" + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century. The name "enneagon" comes from Greek enneagonon (εννεα, nine + γωνον (from γωνία = corner)), and is arguably more correct, though somewhat less common.

A regular nonagon has internal angles of 140°. The area of a regular nonagon of side length a is given by

$A = \frac{9}{4}a^2\cot\frac{\pi}{9}\simeq6.18182\,a^2.$

Construction

Although a regular nonagon is not constructible with compass and straightedge there are methods of construction that produce very close approximations.

Below is a close approximation of a regular nonagon. The approximate angle error is given in the animation.

Graphs

The K₉ complete graph is often drawn as a regular nonagon with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex. Three other rectified forms also project inside of a regular enneagon:

8-simplex (8D)

Rectified 8-simplex (8D)

Birectified 8-simplex (8D)

Trirectified 8-simplex (8D)

Pop culture references

They Might Be Giants have a song entitled "Nonagon" on their children's album "Here Come the 123s." It refers to both an attendee at a party at which "everybody in the party is a many-sided polygon" and a dance they perform at this party.^[1]

See also

• Enneagram (nonagram)

References

1. ^ TMBW.net

• Weisstein, Eric W., "Nonagon" from MathWorld.

External links

• Properties of a Nonagon (with interactive animation)
• Nonagrams