About o Regular Octagon
What Is a Regular Octagon?
A regular octagon is an eight-sided polygon with all sides equal and all angles equal. Each interior angle measures 135 degrees. The octagon is one of the most recognizable geometric shapes, appearing in stop signs, architecture, and design worldwide.
Area Formula
The area of a regular octagon with side length s is A = 2(1 + sqrt(2)) x s². This can also be expressed as A = 2a² x tan(67.5°) or A = 8 x (1/2) x s x a where a is the apothem (inradius).
Properties
A regular octagon has 20 diagonals total. The longest diagonal (connecting opposite vertices) has length d = s x sqrt(4 + 2sqrt(2)). The circumradius R = (s/2) x sqrt(4 + 2sqrt(2)) and the apothem (inradius) r = (s/2) x (1 + sqrt(2)).
Applications
Octagons are used in traffic signs (stop signs), architectural design (Gazebo layouts, floor tiles), watch faces, and umbrella frames. The octagonal shape provides excellent structural stability and efficient space utilization.