hexadecagon

A hexadecagon is a polygon with sixteen sides and sixteen vertices. In its regular form all sides and all interior angles are equal. Regular hexadecagons can be formed by connecting points that are equally spaced around a circle. The figure is constructible with a compass and straight‑edge because 16 is a power of two; this follows from the general fact that regular n‑gons are constructible when n is the product of a power of two and distinct Fermat primes.

The sum of the interior angles of a hexadecagon is (16 − 2) × 180°, or 2520°, so each interior angle of a regular hexadecagon measures 157.5°. The exterior angle, which is supplementary to the interior angle, is 22.5°. A hexadecagon has 16 axes of symmetry; its symmetry group is the dihedral group D₁₆, containing 32 elements that comprise 16 rotations and 16 reflections. The number of diagonals in any hexadecagon is n(n − 3)/2, which for n = 16 gives 104.

If s denotes the side length, the perimeter is 16s. The apothem a, the distance from the centre to the midpoint of a side, is a = s/(2 tan (π/16)). The area can be expressed either as ½ × perimeter × apothem, giving A = 8s² cot(π/16), or by using the circumradius R = s/(2 sin (π/16)), so that A = ½ nR² sin(2π/n). Regular hexadecagons appear in architectural tilings, fractal constructions that approximate circles, and as a test case for generating regular polygons in computer graphics. Overall, the hexadecagon exemplifies the rich symmetry and geometric properties that arise in higher‑order polygons.