Tridecagon

A tridecagon is a polygon with 13 sides. It also known as a triskaidecagon. It has an internal angle degree of ~152.308 degrees and a total of 6840 degrees. The area is $A={\frac{13}{4}} a^{2} \cot {\frac{\pi}{13}}$ which is about $13.1858 a^2$ where $a$ is the side length. This cannot be constructed by using a compass and straightedge, but can be constructed using an angle trisector or neusis. But, what if there's is not a side length value given? There is a formula for that too! The side length, where $r$ is the radius of the circumcircle that this is being constructed on, of a tridecagon is $r \cdot 2 \cdot \sin{\frac{\pi}{13}}$ or $2r \cdot 0.23931566428755777$ . If this was constructed on a unit circle the side length would be $0.478631328575115$ . The error of side length being off is 0.0 for up to 15 decimal places, so pretty accurate. If the radius was 1 billion km, then this formula would be off by lass than 1mm. The central angle of a tridecagon is about $27.6923076923077$ , which is of by $0.0$ degrees up to 13 decimal places. There are exactly $4$ different distinct symmetrical lines on a tridecagon. In everyday life this tridecagon can be seen on a Czech 20 korun coin. A related polygon is a tridecagram which is a 13 sided star shape. The $5$ regular forms are $\frac{13}{2}$ , $\frac{13}{3}$ , $\frac{13}{4}$ , $\frac{13}{5}$ , $\frac{13}{6}$ .

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Tridecagon

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