# Nonagon

(Redirected from Nine-pointed star)
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 /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 is represented by Schläfli symbol {9} and has internal angles of 140°. The area of a regular nonagon of side length a is given by

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

o - circuit

o=9*a

a - length

r - the radius of the inscribed circle

r=(a/2)/tan(π/9)

area:

S=9/2 * a * r = 9 * r² * tan(π/9) = 9/4 * a² /tan(π/9) = 9/4 * a² * cotg(π/9) = 9/2 * R² * sin(2π/9)

R - the radius of the circle circumscribed

R = √( (a/2)² + r² )

R² * cos²(π/9) = r²

## Construction

Although a regular nonagon is not constructible with compass and straightedge there are methods of construction that produce very close approximations. It can be constructed using neusis, or by allowing the use of an angle trisector.

The following is an approximate construction of a nonagon using a straightedge and compass.

Another animation of an approximate construction:

Do downsize the angle 60 ° with four angle bisectors and make a thirds of circular arc MON with an approximate solution between angle bisectors w3 and w4. Straight auxiliary line g aims over the point O to the point N (virtually a ruler at the points O and N applied), between O and N, therefore no auxiliary line.
• Example to illustrating the error: At a circumscribed circle radius r = 100 000 km, the absolute error of the 1st side would be approximately 8,6 mm.

## 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.

 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.