Octagon

An octagon is a polygon with eight sides and eight angles. The name comes from the Greek okto “eight” and gonia “angle.” If all sides and angles are equal, the figure is a regular octagon.

Regular octagon properties include an interior angle of 135 degrees and an exterior angle of 45 degrees. It has eightfold rotational symmetry and eight mirror symmetries. A regular octagon contains 20 diagonals. For a side length a, its perimeter is 8a and its area is 2(1 + √2) a². If the octagon is inscribed in a circle, various trigonometric relationships relate side length, radius, and diagonal lengths.

In practical use, the stop sign is a common example of a regular octagon. Regular octagons appear in architecture and graphic design and in tiling patterns such as the 4.8.8 Archimedean tiling, where octagons meet squares.

Irregular octagons have eight sides that are not all equal. They can be convex or concave. The area of an irregular octagon is computed by the standard polygon area formula (such as the shoelace formula) or by decomposing the shape into simpler parts and summing their areas. The concept of an octagon extends to various applications in geometry, design, and modeling, where eight-sided figures provide a balance of symmetry and practicality.