Type Regular polygon
Edges and vertices 19
Schläfli symbol {19}
Coxeter diagram
Symmetry group Dihedral (D19), order 2×19
Internal angle (degrees) ≈161.052°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry an enneadecagon is a polygon with 19 sides and angles.[1] It is also known as an enneakaidecagon or a nonadecagon.[2]

A regular enneadecagon is represented by Schläfli symbol {19}.

## Regular form

The radius of the circumcircle of the regular enneadecagon with side length t is $R=\frac{t}{2} \csc \frac {180}{19}$ (angle in degrees). The area, where t is the edge length, is $\frac{19}{4}t^2 \cot \frac{\pi}{19} \simeq 28.4652\,t^2.$

### Construction

A regular enneadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector.

An animation of an approximate construction:

• Example illustrating the error: At a radius r = 1 million km, the absolute error of the 1st side would be approximately 4,5 mm.

## Related polygons

A enneadecagram is a 19-sided star polygon. There are 9 regular forms given by Schläfli symbols: {19/2}, {19/3}, {19/4}, {19/5}, {19/6}, {19/7}, {19/8}, and {19/9}.

Picture Interior angle Picture Interior angle {19/2} {19/3} {19/4} {19/5} ≈142.105° ≈123.158° ≈104.211° ≈85.2632° {19/6} {19/7} {19/8} {19/9} ≈66.3158° ≈47.3684° ≈28.4211° ≈9.47368°

### Petrie polygons

The regular enneadecagon is the Petrie polygon for one higher-dimensional polytope, projected in a skew orthogonal projection:

 18-simplex (18D)

## References

1. ^ Borges, Samantha; Morgan, Matthew (2012), Children's Miscellany: Useless Information That's Essential to Know, Chronicle Books, p. 110, ISBN 9781452119731.
2. ^ McKinney, Sueanne; Hinton, KaaVonia (2010), Mathematics in the K-8 Classroom and Library, ABC-CLIO, p. 67, ISBN 9781586835224.