Octagon Calculator

Octagon Calculator. Enter the side length of a regular octagon and get back every key property: area, perimeter, long diagonal, medium diagonal, short diagonal, circumradius, and inradius. The Octagon Calculator uses exact geometric formulas so you can verify dimensions for flooring tiles, stop signs, room layouts, or any octagonal design project. Also try the calculate Annulus Area (A₀), Outer Circle Area (A₁) & Inner Circle Area (A₂) — Annulus.

Results

Area (A)

Perimeter (P)

Long Diagonal (d)

Medium Diagonal (m)

Short Diagonal (s)

Circumradius (R)

Inradius / Apothem (r)

Interior Angle

How many sides does an octagon have?

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°. See also our find Volume of Hollow Cylinder with Hollow Cylinder Calculator.

What is the formula for the area of a regular octagon?

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.

What are the diagonals of a regular octagon?

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).

What is the difference between circumradius and inradius of an octagon?

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.

What is the sum of interior angles of an octagon?

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°.

How do I calculate the area of an octagon with a side of 5 cm?

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

What are real-world examples of octagons?

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.

Does this calculator work for irregular octagons?

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. You might also find our Pythagorean Theorem Calculator useful.