A regular octagon is a polygon with eight equal sides and eight equal angles, each measuring 135°. The term comes from “octo” (eight) and “gonia” (angle). Regular octagons appear frequently in architecture, design, and nature—from stop signs to building floor plans to honeycomb-like patterns. The octagon has remarkable symmetry: it has 8-fold rotational symmetry and 8 lines of reflectional symmetry. When you inscribe a regular octagon in a square by cutting the corners, you create the classic shape seen in stop signs and many architectural applications. The octagon bridges the regularity of a square with the smoothness approaching a circle.

The mathematical properties of an octagon are elegant and systematic. The area formula A = 2(1 + √2)s² shows how the octagon’s area grows with side length. The octagon has two types of diagonals: shorter ones connecting vertices separated by one vertex, and longer ones connecting opposite vertices. Regular octagons tile the plane perfectly when combined with squares, a pattern exploited in Islamic tile work and modern architecture. Understanding octagons helps in solving problems ranging from land surveying to graphic design, and provides intuition for working with regular polygons generally.