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Nonagon

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Regular nonagon
Regular nonagon
Edges and vertices 9
Schläfli symbols {9}
Coxeter–Dynkin diagrams CDW ringCDW 9CDW dot
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.

Complete graph K9

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

Star polygon 9 2Star polygon 9 4Star figure 9-3

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]

References


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

 

External links


ar:تساعي (مضلع) ast:Eneágonu az:Düzgün doqquzbucaqlı cs:Devítiúhelníkgl:Eneágono it:Ennagono ht:Nonagòn hu:Kilencszög nl:Negenhoekno:Enneagon nn:Enneagon pl:Dziewięciokąt foremny pt:Eneágonosr:Деветоугаоsv:Nonagon th:รูปเก้าเหลี่ยม

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