A triacontagon, commonly called a 30-gon, is a polygon with thirty sides and thirty vertices. The convex regular triacontagon is often denoted by the Schläfli symbol {30}; it is one of many polygons studied in planar geometry and can be used as a practical approximation of a circle in design and engineering. For general reference about polygons see related resources.
Basic measures and formulas
The sum of interior angles of any 30-gon equals (30-2)·180° = 5040°. In a regular triacontagon each interior angle measures 168°, and the central angle (subtended at the center by one side) is 360°/30 = 12°. Common geometric relations for a regular 30-gon with side length a and circumradius R include:
- side to radius: a = 2R·sin(π/30)
- apothem (inradius): r = R·cos(π/30)
- area (regular): A = (n·a²)/(4·tan(π/n)) with n=30, equivalently A = (30·a²)/(4·tan(π/30))
Symmetry and star polygons
The full symmetry group of the regular triacontagon is the dihedral group D30 of order 60, consisting of 30 rotations and 30 reflections. Beyond the convex form {30}, there exist several regular star polygons of the form {30/k} for integers k coprime to 30; these produce nonconvex star figures with different vertex connections but the same vertex set.
Constructibility and algebraic facts
Because 30 = 2·3·5 and 3 and 5 are Fermat primes, the regular triacontagon is constructible with straightedge and compass. Angles of 12° (the central angle) and related trigonometric values can therefore be expressed in nested square roots and constructed by classical Euclidean methods.
In practice triacontagons appear in decorative tiling, gear-like approximations of circles, and theoretical studies of polygonal symmetry. Distinctions to note include convex versus star (nonconvex) forms, and the difference between equilateral but non-equiangular 30-gons and the perfectly regular case where both properties hold simultaneously.