In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900ð.
The area of a regular 257-gon is (with )
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 2<sup>2<sup>n</sup></sup> + 1 (in this case n = 3). Thus, the values and are 128-degree algebraic numbers, and like all constructible numbers they can be written using square roots and no higher-order roots.
Although it was known to Gauss by 1801 that the regular 257-gon was constructible, the first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822) and Friedrich Julius Richelot (1832). Another method involves the use of 150 circles, 24 being Carlyle circles: this method is pictured below, along with a full construction showing all steps. One of these Carlyle circles solves the quadratic equation x<sup>2</sup> + x − 64 = 0.
The regular 257-gon has Dih<sub>257</sub> symmetry, order 514. Since 257 is a prime number there is one subgroup with dihedral symmetry: Dih<sub>1</sub>, and 2 cyclic group symmetries: Z<sub>257</sub>, and Z<sub>1</sub>.
A 257-gram is a 257-sided star polygon. As 257 is prime, there are 127 regular forms generated by Schläfli symbols {257/n} for all integers 2 ⤠n ⤠128 as .
Below is a view of {257/128}, with 257 nearly radial edges, with its star vertex internal angles 180ð/257 (~0.7ð).