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Tridecagon

Tridecagon is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Tridecagon rather than just read about it. In short: 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}.

Tridecagon — main illustration
Tridecagon — illustration

Key takeaways

  • Tridecagon belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Tridecagon to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Tridecagon from memory before moving on to harder problems.

Reference excerpt

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

A = 13 4 a 2 cot ⁡ π 13 ≃ 13.1858 a 2 . {\displaystyle A={\frac {13}{4}}a^{2}\cot {\frac {\pi }{13}}\simeq 13.1858\,a^{2}.}

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 O A ¯ = 12 , {\displaystyle {\overline {OA}}=12,} according to Andrew M. Gleason, based on the angle trisection by means of the Tomahawk (light blue).

Symmetry

The regular tridecagon has Dih13 symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z13, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order. 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.

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.

Although 13-sided stars appear in the Topkapı Scroll, they are not of these regular forms.

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

References

External links Weisstein, Eric W. "Tridecagon". MathWorld.

Illustrations

Tridecagon illustration
Tridecagon: A neusis construction of a regular tridecagon (triskaidecagon) with radius of circumcircle O A ¯ = 12 {\displaystyle {\overline {OA}}=12} as an animation (1 min 44 s), angle trisection by means of the Tomahawk (light blue). This construction is derived from the following equation: cos ⁡ ( 2 π 13 ) = 1 12 ( 2 26 − 2 13 cos ⁡ ( 1 3 arctan ⁡ ( 26 + 5 13 9 ) ) + 13 − 1 ) . {\displaystyle \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).}
A neusis construction of a regular tridecagon (triskaidecagon) with radius of circumcircle O A ¯ = 12 {\displaystyle {\overline {OA}}=12} as an animation (1 min 44 s), angle trisection by means of the Tomahawk (light blue). This construction is derived from the following equation: cos ⁡ ( 2 π 13 ) = 1 12 ( 2 26 − 2 13 cos ⁡ ( 1 3 arctan ⁡ ( 26 + 5 13 9 ) ) + 13 − 1 ) . {\displaystyle \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).}
Tridecagon: 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.
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.
Tridecagon illustration
Tridecagon illustration

Worked examples

Example 1 — a first encounter with Tridecagon

Start with the simplest possible case. Write down what Tridecagon claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tridecagon before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tridecagon ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tridecagon

In research
Tridecagon appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Tridecagon in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Tridecagon is common in secondary-school and first-year university syllabi. It links to neighbouring topics Polygons by the number of sides, so understanding it makes those chapters shorter.
In everyday life
Look for Tridecagon outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Tridecagon in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Tridecagon means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Tridecagon out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tridecagon in simple terms?

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}.

Why does Tridecagon matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tridecagon?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tridecagon.

Tags

  • Polygons by the number of sides

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