65537-gon

Regular 65537-gon
A regular 65537-gon
Type	Regular polygon
Edges and vertices	65537
Schläfli symbol	{65537}
Coxeter diagram
Symmetry group	Dihedral (D65537), order 2×65537
Internal angle (degrees)	≈179.99°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a 65537-gon is a polygon with 65537 sides. The sum of the interior angles of any non-self-intersecting 65537-gon is 11796300°.

The area of a regular 65537-gon is (with t = edge length)

A whole regular 65537-gon is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 15 parts per billion.

The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65537 is a Fermat prime, being of the form 2²ⁿ + 1 (in this case n = 4). Thus, the values ${\displaystyle \cos {\frac {\pi }{65537}}}$ and ${\displaystyle \cos {\frac {2\pi }{65537}}}$ are of a 32768-degree algebraic numbers, and like any constructible numbers they can be written in terms of square roots and no higher-order roots.