octagonal

Octagonal is the adjective used for things that have eight sides or eight angles. In geometry, an octagon is a polygon with eight sides and eight interior angles; the sum of those angles is 1080 degrees. A regular octagon has all sides equal and all interior angles equal to 135 degrees. For a regular octagon with side length s, the area is A = 2(1+√2)s^2.

The term derives from Latin octagonum, from Greek okto- eight and gonia-angle. Octagonal shapes occur in many contexts beyond basic geometry, including regular octagons, irregular octagons, and star polygons such as the octagram (a star polygon derived from an eight-point figure).

In architecture, design, and art, octagonal plans and features are used for aesthetic and functional reasons, including rooms, towers, domes, and pavilions. In tiling and polyhedral studies, octagons appear in mixtures with other polygons, such as the semiregular tiling that uses squares and octagons in a 4.8.8 vertex configuration.

In mathematics, octagonal numbers are a class of figurate numbers. The nth octagonal number is given by n(3n−2), describing a pattern of centered octagonal arrangements. In three dimensions, octagonal prisms have two parallel octagonal bases connected by rectangular faces, and octagonal pyramids have an octagonal base with eight triangular faces meeting at a apex.

Overall, octagonal denotes eight-sidedness across geometry, mathematics, architecture, and related design fields, with regular octagons offering a classic, symmetric example.