What Is an Octagon? Definition and Key Properties
An octagon is a polygon with exactly 8 sides and 8 angles. The word "octagon" comes from the Greek okto (eight) and gonia (angle). When all sides are equal and all interior angles are equal, we call it a regular octagon — the shape this calculator focuses on.
Octagon Angles
The sum of all interior angles of any octagon is always 1080°. In a regular octagon, each interior angle measures exactly:
1080° ÷ 8 = 135°
Each exterior angle of a regular octagon is 45°, and the central angle (from centre to two adjacent vertices) is also 45°.
Area of a Regular Octagon
Given a side length a, the area formula is:
A = 2(1 + √2) × a²
For example, with a = 5 cm: A = 2 × (1 + 1.41421) × 25 ≈ 120.71 cm².
Diagonals of a Regular Octagon
A regular octagon has 20 diagonals in total. They fall into three distinct length groups depending on how many vertices they skip:
- Short diagonal (d₁) — skips 1 vertex: d₁ = a × √(2 + √2) ≈ 1.8478a
- Medium diagonal (d₂) — skips 2 vertices: d₂ = a × (1 + √2) ≈ 2.4142a
- Long diagonal / diameter (d₃) — connects opposite vertices: d₃ = a × √(4 + 2√2) ≈ 2.6131a
Circumradius and Inradius
- Circumradius (R) — radius of the circle passing through all vertices: R = a/2 × √(4 + 2√2) ≈ 1.3066a
- Inradius / Apothem (r) — radius of the inscribed circle (centre to midpoint of a side): r = a(1 + √2)/2 ≈ 1.2071a
How to Use This Octagon Calculator
- Enter the side length in the input field.
- Choose your unit — metric (mm, cm, m, km) or imperial (in, ft, yd, mi).
- Click Calculate to instantly get the area, perimeter, circumradius, inradius, and all three diagonal lengths.
- Use the share button to save or send your result via a unique link.
How to Draw a Regular Octagon
The simplest method is to start with a square, then cut the four corners at 45° at a length of a/√2 from each corner, where a is the desired side length. This gives you eight equal sides meeting at 135° angles. Alternatively, inscribe it in a circle and mark 8 equally-spaced points (every 45°).
Octagons in Real Life
- Stop signs — the most globally recognised octagonal object, chosen for its unique shape so drivers can identify it even when obscured.
- Octagon house — a 19th-century architectural style popularised by Orson Squire Fowler, designed to maximise interior space and natural light.
- Octagonal tiles — a classic flooring pattern, often combined with small square infill tiles, found in Victorian and Art Deco interiors.
- Camera apertures — many camera lenses create octagonal bokeh because the iris diaphragm has 8 blades.
- UFC Octagon — the iconic 8-sided fighting cage used in mixed martial arts competitions.
- Bathroom and kitchen tiles — octagon tiles paired with dot tiles create visually striking, space-efficient patterns.
Summary of Formulas
| Property | Formula |
|---|---|
| Perimeter | P = 8a |
| Area | A = 2(1 + √2) × a² |
| Circumradius | R = a/2 × √(4 + 2√2) |
| Inradius | r = a(1 + √2) / 2 |
| Short diagonal | d₁ = a × √(2 + √2) |
| Medium diagonal | d₂ = a × (1 + √2) |
| Long diagonal | d₃ = a × √(4 + 2√2) |
| Interior angle | 135° |