Octagon calculator
This octagon calculator turns one known value of a regular octagon – side, width across flats, width across corners, area or perimeter – into every other dimension.
Example: Known value: Side length a · Value 10 · Unit: cm → Result: Side 10 cm: area 482.8427 cm², width across flats 24.1421 cm. Source: Wolfram MathWorld, Regular Octagon: inradius r = ½(1+√2)·a, circumradius R = ½√(4+2√2)·a, area A = 2(1+√2)·a², interior angle 3π/4 = 135° (read 14.10.2026). Updated: .
How it is calculated
The area of a regular octagon is A = 2(1 + √2)·a² ≈ 4.8284·a². A side of 10 cm gives about 482.8427 cm² and a width across flats of 24.1421 cm.
Octagon formulas
- Perimeter: P = 8a
- Width across flats (distance between opposite sides): W = (1 + √2)·a ≈ 2.4142·a
- Width across corners: D = √(4 + 2√2)·a ≈ 2.6131·a
- Short diagonal (skipping one corner): √(2 + √2)·a ≈ 1.8478·a
- Inradius r = W/2 ≈ 1.2071·a, circumradius R = D/2 ≈ 1.3066·a
- Each interior angle is 135°. When it is built from eight boards, each end is cut at 22.5°.
For woodwork the width is often known first: a table 60 cm across flats needs sides of 60 ÷ 2.4142 ≈ 24.8528 cm and covers about 2,982.3376 cm².
Examples
| Side a | Area | Across flats | Across corners | Perimeter |
|---|---|---|---|---|
| 1 | 4.8284 | 2.4142 | 2.6131 | 8 |
| 5 | 120.7107 | 12.0711 | 13.0656 | 40 |
| 10 | 482.8427 | 24.1421 | 26.1313 | 80 |
| 30 | 4,345.5844 | 72.4264 | 78.3938 | 240 |
For the saw settings see the miter angle calculator; for six sides the hexagon calculator, for any number of sides the regular polygon calculator.
Limits
The formulas hold for a regular octagon only. A stretched octagon (for example a rectangle with cut corners) has two different side lengths and needs its own measurements.
Frequently asked questions
How do you calculate the area of an octagon?
For a regular octagon: A = 2(1 + √2)·a² ≈ 4.8284·a². A side of 5 gives about 120.7107.
How do I get the side length from the width?
Divide the width across flats by 1 + √2 ≈ 2.4142. A 60 cm octagon has sides of about 24.8528 cm.
What angle do I cut for an octagon?
Each interior angle is 135°, so eight pieces are joined with a 22.5° cut on each end.
What is the difference between across flats and across corners?
Across flats is the distance between two opposite sides (2.4142·a), across corners the distance between two opposite corners (2.6131·a).
What is the perimeter of an octagon?
P = 8a. Sides of 10 cm give a perimeter of 80 cm.
Sources and legal basis
- Wolfram MathWorld, Regular Octagon: inradius r = ½(1+√2)·a, circumradius R = ½√(4+2√2)·a, area A = 2(1+√2)·a², interior angle 3π/4 = 135° (read 14.10.2026)
- Wolfram MathWorld, Regular Polygon: interior angle (n−2)·180°/n, inradius ½·a·cot(π/n), circumradius ½·a·csc(π/n) (read 14.10.2026)
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