Overview

A tetracontagon, commonly called a 40-gon, is a polygon with forty straight sides and forty vertices. The name combines the Greek prefix "tetraconta-" (forty) with "-gon" (angle). The regular tetracontagon has equal sides and equal angles and is a useful theoretical example when studying polygonal approximations of the circle and symmetry groups. For a general introduction see related geometry resources.

Basic properties

Key numerical facts for any tetracontagon include the sum of interior angles and the number of diagonals. The sum of interior angles is (40-2)·180° = 6,840°. Each interior angle of a regular tetracontagon measures 171° and each exterior angle is 9°. The number of distinct diagonals is n(n−3)/2 = 40·37/2 = 740.

Regular tetracontagon: formulas and symmetry

The regular 40-gon is denoted {40} in Schläfli notation. Side length s and circumradius R are related by s = 2R·sin(π/40). The area of a regular tetracontagon with side s can be written as A = (40·s²)/(4·tan(π/40)) = 10·s²/ tan(π/40). Its full symmetry group is the dihedral group D40 of order 80, with a cyclic rotation subgroup C40.

Constructibility and star polygons

By the classical criterion of Gauss and Wantzel, a regular n-gon is constructible with straightedge and compass precisely when n = 2^k times a product of distinct Fermat primes. Because 40 = 2^3·5 and 5 is a Fermat prime, the regular tetracontagon is constructible. Regular star polygons also exist on the same 40 equally spaced vertices; common examples are {40/3}, {40/7} and other {40/k} where k is coprime to 40, producing a family of star figures with their own visual and combinatorial properties.

Uses and notable facts

  • Mathematical contexts: serves as an instructive example in polygon formulae, symmetry studies and constructibility proofs.
  • Design and modelling: large-sided regular polygons approximate circles and appear in tiling patterns, decorative motifs and computational geometry.
  • Distinction: an irregular tetracontagon has sides or angles that vary; only the regular form enjoys equal-angle properties, simple formulae and full dihedral symmetry.

The tetracontagon bridges elementary polygon geometry and more advanced topics like algebraic constructibility and group symmetry, making it a useful case study despite being uncommon in everyday objects.