Octagon Calculator

A regular octagon features eight equal sides and internal angles of 135 degrees each. Because all regular octagons are geometrically similar, knowing just one primary dimension—such as the side length (s)—allows you to derive every other attribute using established trigonometric and algebraic formulas. The area is computed using the formula Area = 2 * s^2 * (1 + sqrt(2)), while the perimeter is simply P = 8 * s. Diagonals and radii (such as the circumcircle and incircle radii) scale proportionally based on factors involving square roots of two, allowing seamless conversion between any input variables.

Let us walk through a manual calculation for a regular octagon with a side length (s) of 5 cm. First, find the perimeter by multiplying the side length by eight: P = 8 * 5 = 40 cm. Next, calculate the area using the standard regular polygon formula: Area = 2 * (5^2) * (1 + sqrt(2)) = 2 * 25 * (1 + 1.4142) = 50 * 2.4142 = 120.71 square centimeters. Finally, you can determine the longest diagonal using the multiplier s * sqrt(4 + 2 * sqrt(2)), which yields approximately 5 * 2.613 = 13.07 cm, giving you a complete profile of the shape.

When designing physical objects or architectural elements with octagonal footprints, always verify whether your given measurements refer to the side length, edge-to-edge distance (incircle diameter), or vertex-to-vertex distance (circumcircle diameter). Confusing the incircle radius with the circumcircle radius is the most common pitfall in polygon calculations. Additionally, double-check your unit consistency; mixing centimeters and meters will drastically skew your area and perimeter outputs.