Quinquatrigintagons

A quinquatrigintagon is a polygon with fifty-three sides and fifty-three vertices. The term derives from the Latin words *quinquaginta* (meaning fifty) and *triginta* (meaning thirty), combined with the Greek suffix *-gon*, which refers to a polygon. This shape belongs to the broader category of regular polygons, which are defined by equal side lengths and equal angles between adjacent sides.

Regular quinquatrigintagons possess a high degree of symmetry, with rotational symmetry of order fifty-three and dihedral symmetry of order one hundred and six. Each interior angle of a regular quinquatrigintagon measures approximately 169.06 degrees, calculated using the formula for the interior angle of a regular *n*-sided polygon: ((*n* − 2) × 180°) / *n*. Due to their large number of sides, quinquatrigintagons closely approximate a circle, making them useful in geometric approximations and mathematical modeling.

Constructing a regular quinquatrigintagon with straightedge and compass alone is theoretically possible but impractical due to its complexity. The construction relies on the exact divisibility of the circle into fifty-three equal parts, which requires solving a trigonometric equation that cannot be expressed using basic geometric constructions. However, numerical methods or computational tools can approximate the shape with high precision.

In geometry and mathematics, quinquatrigintagons serve as examples of high-order polygons, illustrating concepts such as side length, angle measurement, and symmetry. They are rarely encountered in practical applications but remain significant in theoretical studies of polygon properties and their convergence toward circular forms as the number of sides increases.