Octagons

An octagon is a polygon with eight sides and eight vertices. The name comes from the Greek okto (eight) and gonia (angle). In any polygon, the sum of interior angles is (n − 2) × 180 degrees; for an octagon that equals 1080 degrees. A regular octagon has all sides equal and all interior angles equal to 135 degrees.

A regular octagon has eight diagonals from vertex to nonadjacent vertex, and the total number of diagonals is 8 × 5 / 2 = 20. Its symmetry group is the dihedral group D8, comprising 8 rotations and 8 reflections (a total of 16 symmetries).

For a regular octagon with side length a, the perimeter is P = 8a, and the area is A = 2(1 + √2) a^2. The area can also be expressed in terms of the circumradius R as A = 2√2 R^2.

Octagons occur in various contexts, including architecture and design. The stop sign is a common real-world example of a regular octagon. In tiling, octagons can combine with squares to form uniform patterns, such as the 4.8.8 tiling, illustrating how octagons contribute to geometric mosaics.