Tridecagon (also spelled triskaidecagon, tridecagon, trisdecagon or commonly called a 13‑gon) denotes any polygon with 13 straight sides and 13 vertices. The term combines Greek numeric roots for thirteen and the suffix "-gon" for angle. Regular tridecagons have all sides and interior angles equal; irregular tridecagons vary in side length and angle measures.

Basic properties

For a regular tridecagon with side length a, each interior angle measures (n−2)×180°/n = 1980°/13 ≈ 152.3077°. The corresponding exterior angle is about 27.6923°. The area of a regular form can be written as A = (13/4)a² cot(π/13), where cot denotes the cotangent function. Symmetry of the regular tridecagon is the dihedral group D13 of order 26.

The regular 13‑gon is not constructible with only straightedge and compass because 13 is not a Fermat prime; its coordinates require solving higher‑degree cyclotomic equations. There are also several regular star polygons formed from the same 13 vertices, commonly denoted {13/2}, {13/3}, {13/4}, {13/5} and {13/6}, giving distinct star shapes when vertices are connected by skipping vertices.

Appearance and uses

Tridecagons occur primarily in decorative design, heraldry, tiling motifs, digital icons and mathematical illustrations rather than in common architectural modules. Regular and starred 13‑gon motifs are used where a visually balanced but uncommon symmetry is desired. Because 13 is culturally significant in many societies, thirteen‑sided outlines can carry symbolic meaning in art and design.

Names and cultural notes

  • Common names: tridecagon, triskaidecagon, trisdecagon, 13‑gon.
  • Etymology: mixes Greek elements for three and ten (tri‑ + deca‑) or the Greek word triskaideka (thirteen).
  • Notable fact: there are six distinct regular star forms for 13‑sided arrangements, corresponding to different step sizes.

For a concise visual reference or more technical details about polygon notation and star polygons see this entry.