Nonagon

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Regular nonagon (enneagon)
Regular polygon 9 annotated.svg
A regular nonagon (enneagon)
Type Regular polygon
Edges and vertices 9
Schläfli symbol {9}
Coxeter diagram CDel node 1.pngCDel 9.pngCDel node.png
Symmetry group Dihedral (D9), order 2×9
Internal angle (degrees) 140°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a nonagon /ˈnɒnəɡɒn/ (or enneagon /ˈɛnəɡɒn/) 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[edit]

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.

Approximated Nonagon Inscribed in a Circle.gif

Graphs[edit]

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 t0.svg
8-simplex (8D)
8-simplex t1.svg
Rectified 8-simplex (8D)
8-simplex t2.svg
Birectified 8-simplex (8D)
8-simplex t3.svg
Trirectified 8-simplex (8D)

Pop culture references[edit]

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[edit]

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

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

See also[edit]

References[edit]

  1. ^ TMBW.net

External links[edit]