Nonagon

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Regular nonagon
Edges and vertices 9
Schläfli symbols {9}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D9)
Area
(with t=edge length
$A = \frac{9}{4}t^2\cos\frac{\pi}{9}$$\simeq6.18182t^2$
Internal angle
(degrees)
140°

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

The name "nonagon" is a 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

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

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

Graphs and stars

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.

The regular nonagon also contains 3 star figures, the third being the compound of three equilateral triangles:

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]

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)