Hexadecagons

A hexadecagon is a polygon with sixteen sides and sixteen vertices. The name derives from the Greek words hexadeka, meaning sixteen, and gonia, meaning angle or corner. In general usage the term “sixteen-gon” is also common, while “hexakaidecagon” is another Greek-based form.

A regular hexadecagon has all sides equal and all interior angles equal. The interior angle measures 157.5 degrees (each exterior angle is 22.5 degrees). It is cyclic, meaning all vertices lie on a common circle, with a central angle of 22.5 degrees subtending each side. If the polygon has side length s and circumradius R, relationships include s = 2R sin(π/16) and the area A = (n s^2) / (4 tan(π/n)) = 4 s^2 cot(π/16). Equivalently, A can be expressed in terms of R as A = (n/2) R^2 sin(2π/n).

Diagonals: a hexadecagon has 104 diagonals. The number of diagonals in a regular n-gon is n(n−3)/2, which yields 16·13/2 = 104.

Construction: a regular hexadecagon is constructible with straightedge and compass since 16 = 2^4. A common method is to start with a circle, construct a regular octagon, and then bisect each central angle to obtain a sixteen-sided figure inscribed in the same circle.

Applications and occurrence: hexadecagons appear mainly in theoretical geometry and mathematical problems rather than everyday designs. They serve as a canonical example of a highly symmetric, even-sided polygon and illustrate concepts of regular polygons, diagonals, and polygon area formulas.