Octagon Calculator
Calculate all properties of a regular octagon from its side length. This calculator instantly computes the area, perimeter, apothem, and longest diagonal of any regular octagon.
A regular octagon has eight equal sides and eight equal interior angles of 135 degrees each. It is one of the most recognizable geometric shapes, most famously used for stop signs worldwide.
Regular Octagon Formulas (side length = s):
- Area: A = 2(1 + sqrt(2)) x s^2, which is approximately 4.8284 x s^2
- Perimeter: P = 8s
- Apothem: a = (s/2)(1 + sqrt(2))
- Longest Diagonal: d = s x sqrt(4 + 2 x sqrt(2))
The area formula can be derived by dividing the octagon into a central square, four rectangles, and four corner triangles, or equivalently into eight isosceles triangles from the center.
Regular octagons appear in architecture (baptisteries, towers), urban planning (intersections), floor tiling (often combined with squares), engineering (nuts and socket heads), and of course traffic signs. The octagon's near-circular shape provides good area-to-perimeter efficiency while being easier to construct than a true circle.