octogons

Octogons, or octagons, are polygons with eight sides and eight vertices. The standard spelling is octagon, and octagons is the common plural; octogons is another historical or less common variant. The name derives from the Greek oktṓ (eight) and gōnía (angle).

A polygon with eight sides has an interior angle sum of (8−2)×180 = 1080 degrees. In a regular octagon, all sides and all interior angles are equal, giving each interior angle of 135 degrees and each exterior angle of 45 degrees. The number of diagonals is 20 (calculated by n(n−3)/2 with n = 8).

For a regular octagon with side length a, the area is A = 2(1+√2)a^2. An easy construction is to truncate the corners of a square; removing isosceles right triangles from the corners leaves a regular octagon. A regular octagon can also be inscribed in a circle; the side length relates to the circumradius by s = R√(2−√2).

Octogons can be convex or concave; the term octagon typically refers to convex shapes unless stated otherwise. In architecture and signage, regular octagons are notable for the stop sign, which uses a red stop octagon. In tiling, regular octagons appear in Archimedean tilings together with squares, creating a repeating pattern.

See also: octagon, polygon, stop sign, Archimedean tiling.