{{short description|Polygon with 13 edges}} {{Expand German|topic=sci|date=October 2025}} {{Regular polygon db|Regular polygon stat table|p13}} In geometry, a '''tridecagon''' or '''triskaidecagon''' or 13-gon is a thirteen-sided polygon.

== Regular tridecagon == A ''regular tridecagon'' is represented by Schläfli symbol {13}.

The measure of each internal angle of a regular tridecagon is approximately 152.308 degrees, and the area with side length ''a'' is given by : <math>A = \frac{13}{4}a^2 \cot \frac{\pi}{13} \simeq 13.1858\,a^2.</math>

== Construction == As 13 is a Pierpont prime but not a Fermat prime, the regular tridecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or angle trisection.

The following is an animation from a ''neusis construction'' of a regular tridecagon with radius of circumcircle <math>\overline{OA} = 12,</math> according to Andrew M. Gleason,<ref name=Gleason>{{cite journal|last=Gleason|first=Andrew Mattei|title=Angle trisection, the heptagon, and the triskaidecagon p.&nbsp;192–194 (p. 193 Fig.4)|journal=The American Mathematical Monthly|date=March 1988|volume=95|issue=3 |pages=186–194|url=http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#10 |archive-url=https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf#10 |access-date=24 December 2015|doi= 10.2307/2323624|archive-date=2015-12-19 |url-status=dead}}</ref> based on the angle trisection by means of the Tomahawk (light blue). left|700px|thumb|A neusis construction of a regular tridecagon (triskaidecagon) with radius of circumcircle <math>\overline{OA} = 12</math> as an animation (1&nbsp;min&nbsp;44 s), angle trisection by means of the Tomahawk (light blue). This construction is derived from the following equation:<br /> <math>\cos\left(\frac{2\pi}{13}\right)=\frac{1}{12}\left(2\sqrt{26-2\sqrt{13}}\cos\left(\frac{1}{3}\arctan\left(\frac{26+5\sqrt{13}}{9}\right)\right)+\sqrt{13}-1\right).</math> {{clear}}

== Symmetry == thumb|200px|Symmetries of a regular tridecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices and edge. Gyration orders are given in the center.

The ''regular tridecagon'' has Dih<sub>13</sub> symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih<sub>1</sub>, and 2 cyclic group symmetries: Z<sub>13</sub>, and Z<sub>1</sub>.

These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, {{isbn|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp.&nbsp;275–278)</ref> Full symmetry of the regular form is '''r26''' and no symmetry is labeled '''a1'''. The dihedral symmetries are divided depending on whether they pass through vertices ('''d''' for diagonal) or edges ('''p''' for perpendiculars), and '''i''' when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as '''g''' for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the '''g13''' subgroup has no degrees of freedom but can be seen as directed edges.

== Numismatic use == The regular tridecagon is used as the shape of the Czech 20 korun coin.<ref>Colin R. Bruce, II, George Cuhaj, and Thomas Michael, ''2007 Standard Catalog of World Coins'', Krause Publications, 2006, {{isbn|0896894290}}, p.&nbsp;81.</ref> : File:20 CZK.png

== Related polygons == A '''tridecagram''' is a 13-sided star polygon. There are 5 regular forms given by Schläfli symbols: {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}. Since 13 is prime, none of the tridecagrams are compound figures.

{| class="wikitable collapsible collapsed" |- ! colspan="12" | Tridecagrams |- style="text-align: center;" ! Picture | 120px<br />{13/2} | 120px<br />{13/3} | 120px<br />{13/4} | 120px<br />{13/5} | 120px<br />{13/6} |- style="text-align: center;" ! Internal angle | ≈124.615° || ≈96.9231° || ≈69.2308° || ≈41.5385° || ≈13.8462° |}

Although 13-sided stars appear in the Topkapı Scroll, they are not of these regular forms.<ref>{{cite journal | last = Cromwell | first = Peter R. | doi = 10.1080/17513470903311669 | issue = 2 | journal = Journal of Mathematics and the Arts | mr = 2786387 | pages = 73–85 | title = Islamic geometric designs from the Topkapı Scroll I: unusual arrangements of stars | url = https://scholar.archive.org/work/ia2j6t4345c67mvlciiihevxze | volume = 4 | year = 2010}}</ref>

=== Petrie polygons === The regular tridecagon is the Petrie polygon of the 12-simplex:

{| class=wikitable |- style="text-align: center;" ! A<sub>12</sub> |- | 125px<br>12-simplex |}

== References == {{reflist}}

== External links == * {{MathWorld |title=Tridecagon |urlname=Tridecagon}} {{Polygons}}

Category:Polygons by the number of sides