Hexacontagon (60-sided polygon)

Geometry and key formulas

Important numerical properties of a regular hexacontagon include:

• Each interior angle equals (n-2)·180°/n = 174°.
• The sum of all interior angles is (n-2)·180° = 10,440°.
• The exterior angle (central rotation step for adjacent vertices) is 6°.
• Number of diagonals: n(n-3)/2 = 1,710.

Area formulas for a regular 60-gon with side length s or circumradius R are standard for regular n-gons: A = (n s^2)/(4 tan(π/n)) or A = (n/2) R^2 sin(2π/n). Coordinates of vertices on the unit circle can be written as complex numbers e^{2πik/60} for k = 0,1,...,59.

Symmetry and constructibility

The symmetry group of a regular hexacontagon is the dihedral group D60 of order 120, consisting of 60 rotations and 60 reflections. The regular 60-gon is constructible with straightedge and compass because 60 = 2^2·3·5 and the prime factors 3 and 5 are Fermat primes; consequently a sequence of classical constructions can produce its vertices (for example by successive angle divisions to obtain 6° steps).

Variations and star polygons

Beyond the regular convex form, there are many star polygon versions denoted {60/k} for integers k that are relatively prime to 60. These stellations produce intricate star shapes by connecting each vertex to the k-th vertex around the circle. The family of such star figures illustrates how a single vertex set can produce multiple distinct polygonal circuits.

History, name, and uses

Polygons with many sides have been studied since antiquity in the context of circle approximation, geometry, and construction problems. The specific prefix "hexacont-" derives from classical Greek numerals. Practical uses of polygons with many sides include geometric modeling, graphic design, tiling motifs, and approximating circular arcs in engineering drawings. For more general background on polygons, see related resources.

Notable facts

Because its exterior angle is exactly 6°, the regular hexacontagon relates naturally to divisions of a circle into 60 parts—a division that also underlies the 60-minute hour and 360-degree circle in historical sexagesimal systems. The large number of diagonals and the abundance of possible star forms make the 60-gon a rich example in studies of polygonal combinatorics and symmetry.