Nonagon

Regular nonagon (enneagon)

A regular nonagon (enneagon)
Type Regular polygon
Edges and vertices 9
Schläfli symbol {9}
Coxeter diagram
Symmetry group Dihedral (D9), order 2×9
Internal angle (degrees) 140°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a nonagon (or enneagon ) 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 than "nonagon".

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 enneagon 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] Slipknot's logo is also a version of a nonagon, being a nine-pointed star made of 3 triangles.

Architecture

Temples of Baha'i Faith are required to be nonagonal.

The U.S. Steel Tower is an irregular nonagon.