Pentacontagon (50-sided polygon)

Geometric characteristics

For a regular pentacontagon (all sides and angles equal) simple numerical values follow directly from n = 50. The central angle between consecutive vertices is 360°/50 = 7.2°. Each interior angle measures 180° - 360°/50 = 172.8°, so the polygon's edges deviate only slightly from a straight line. Triangulation of a pentacontagon produces 48 triangles, and the number of diagonals is 50·47/2 = 1175.

Formulas and algebraic description

• Interior angle sum: (50-2)·180° = 8640°.
• Interior angle (regular): 172.8°; central angle: 7.2°.
• Area for regular pentacontagon with side length s: A = (50/4)·s²·cot(π/50) = (25/2)·s²·cot(π/50), or equivalently A = 25·s·a where a is the apothem.
• Apothem a relates to side s by a = s/(2·tan(π/50)).
• Symmetry of the regular form: dihedral group D50 of order 100.

As with any regular n-gon, the coordinates of its vertices on the unit circle are given by the 50th roots of unity, e^{2πim/50} for m = 0,1,…,49, which connects the pentacontagon to complex numbers and cyclotomic theory.

Constructibility and related figures

The regular pentacontagon has Schläfli symbol {50}. A classical result about straightedge-and-compass constructibility states that a regular n-gon is constructible if and only if n = 2^k times a product of distinct Fermat primes. Because 50 = 2·5^2 contains a repeated Fermat prime (5 appears twice), the regular pentacontagon is not constructible with only compass and straightedge. It can, however, be studied algebraically or approximated and can be drawn by other methods.

Related star polygons are denoted {50/k} for integers k with 2 ≤ k ≤ 24; when gcd(50,k) > 1 the figure decomposes into several identical polygons (a compound), otherwise it produces a single star polygon. These variations are commonly explored in the study of polygonal symmetries and star figure classification.

Etymologically, the name derives from Greek roots: penta- (five) combined with konta (tens) to indicate fifty; alternative spellings reflect different transliterations. In practice, pentacontagons are most often of theoretical and pedagogical interest—used to illustrate polygon formulas, symmetry groups, and relationships among roots of unity—rather than appearing frequently in standard architectural patterns, although any many-sided figure can serve decorative or approximation-to-circle purposes.