257-gon

Regular 257-gon
A regular 257-gon
Type	Regular polygon
Edges and vertices	257
Schläfli symbol	{257}
Coxeter diagram
Symmetry group	Dihedral (D257), order 2×257
Internal angle (degrees)	≈178.60°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a 257-gon (diacosipentacontaheptagon, diacosipentecontaheptagon) is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 91800°.

The area of a regular 257-gon is (with t = edge length)

A whole regular 257-gon is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 24 parts per million.

The regular 257-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 257 is a Fermat prime, being of the form 2²ⁿ + 1 (in this case n = 3). Thus, the values ${\displaystyle \cos {\frac {\pi }{257}}}$ and ${\displaystyle \cos {\frac {2\pi }{257}}}$ are 128-degree algebraic numbers, and like all constructible numbers they can be written using square roots and no higher-order roots.