Hexadecagonal

Hexadecagonal is an adjective relating to a hexadecagon, a polygon with sixteen sides. The term comes from the Greek hexadeca- meaning sixteen and gonia meaning angle, and it is used to describe sixteen-sided figures or properties derived from them.

In geometry, a hexadecagon may be regular or irregular. A regular hexadecagon has all sides and angles equal. Its interior angle is 157.5 degrees, and the sum of interior angles is 2520 degrees. The exterior angle is 22.5 degrees. The figure is constructible with compass and straightedge because sixteen is a power of two.

For a regular hexadecagon with circumradius R, the side length is s = 2R sin(pi/16), and the apothem is a = R cos(pi/16). The area can be expressed as A = (n s^2) / (4 tan(pi/n)); for n = 16 this simplifies to A = 4 s^2 cot(pi/16). These relations allow computation of dimensions and area from a given radius or side length.

Hexadecagons also appear in applied contexts. A regular hexadecagon cannot tile the plane by itself, but it may participate in tilings with other polygons. In three dimensions, hexadecagonal prisms use a hexadecagonal base. In number theory, hexadecagonal numbers form the sequence P(16,n) = 7n^2 - 6n, starting 1, 16, 45, 88, 145, 216, and so on.

The term hexadecagonal is also encountered in crystallography and geometry to describe 16-fold symmetries and related patterns in certain tilings and materials.