|Regular nonagon (enneagon)|
A regular nonagon (enneagon)
|Edges and vertices||9|
|Symmetry group||Dihedral (D9), order 2×9|
|Internal angle (degrees)||140°|
|Properties||convex, cyclic, equilateral, isogonal, isotoxal|
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".
o - circuit
a - length
r - the radius of the inscribed circle
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
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.
Another animation of an approximate construction
Downsize the angle JMK (also 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. Thus, the circular arc MON is freely accessible for the later intersection point R.
RMK = 40.0000000052441...° ; 360° : 9 = 40° ; RMK - 40° = 5.2...E-9°
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.
See also the calculation (Berechnung).
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. Slipknot's logo is also a version of a nonagon, being a nine-pointed star made of 3 triangles.
Temples of Baha'i Faith are required to be nonagonal.
The U.S. Steel Tower is an irregular nonagon.
- Properties of a Nonagon (with interactive animation)