Enneadecagon

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Regular enneadecagon
Regular polygon 19 annotated.svg
A regular enneadecagon
Type Regular polygon
Edges and vertices 19
Schläfli symbol {19}
Coxeter diagram CDel node 1.pngCDel 19.pngCDel node.png
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[edit]

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[edit]

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

Related polygons[edit]

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

Petrie polygons[edit]

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

18-simplex t0.svg
18-simplex (18D)

References[edit]

  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 .

External links[edit]