Octagon Calculator

An octagon has exactly 8 sides. The word comes from the Greek 'okto' (eight) and 'gonia' (angle). A regular octagon has all 8 sides equal in length and all 8 interior angles equal to 135°.

The area of a regular octagon with side length a is A = 2 × (1 + √2) × a². For example, a regular octagon with a side of 5 cm has an area of 2 × (1 + 1.41421) × 25 ≈ 120.71 cm².

A regular octagon has three distinct diagonal lengths. The long diagonal (d) connects opposite vertices and equals a × √(4 + 2√2). The medium diagonal (m) equals a × (1 + √2). The short diagonal (s) connects vertices two apart and equals a × √(2 + √2).

The circumradius (R) is the radius of the circle that passes through all 8 vertices of the octagon, equal to a × √(2 + √2) / √2. The inradius (r), also called the apothem, is the radius of the largest circle that fits inside the octagon and touches all 8 sides, equal to a × (1 + √2) / 2.

The sum of all interior angles of any octagon is 1080°. This is calculated using the formula (n − 2) × 180°, where n = 8. In a regular octagon, each interior angle measures 1080° ÷ 8 = 135°.

Using the formula A = 2 × (1 + √2) × a², substitute a = 5: A = 2 × (1 + 1.41421) × 25 = 2 × 2.41421 × 25 ≈ 120.71 cm².

Stop signs are the most recognizable regular octagon in everyday life. Other examples include octagonal floor tiles, certain clock faces, the famous Octagon House in Washington D.C., UFC/MMA fighting rings, and camera lens aperture designs. Octagonal columns and windows also appear frequently in architecture.

No — this calculator is designed for regular octagons, where all 8 sides are equal and all interior angles are 135°. Irregular octagons (with unequal sides or angles) require much more complex calculations that depend on the specific coordinates or measurements of each side.