Octagon Calculator
Solve octagon area, perimeter, side length, inradius, and circumradius.
Calculator
Results
Side length
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Perimeter
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Area
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Inradius (apothem)
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Circumradius
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Square side (if cut)
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Steps
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Regular octagon formulas
For a regular octagon with side \( s \):
Area: \( A = 2(1 + \sqrt{2})s^2 \)
Perimeter: \( P = 8s \)
Inradius (apothem): \( r = \frac{s}{2 \tan(\pi/8)} \)
Circumradius: \( R = \frac{s}{2 \sin(\pi/8)} \)
Numerically, \( 2(1 + \sqrt{2}) \approx 4.82842712 \), so \( A \approx 4.82842712 s^2 \).
Octagon obtained by cutting the corners of a square
If you start from a square of side \( S \) and cut equal isosceles right triangles from each corner so that a regular octagon remains, the octagon side is:
\( s = S(\sqrt{2} - 1) \)
and so \( S = \dfrac{s}{\sqrt{2} - 1} \)
FAQ
Is this only for regular octagons?
Yes. Irregular octagons need coordinates or side-angle data; this tool follows the regular case like the common online calculators.
What unit should I use?
Use whatever you measure in (mm, cm, m, in, ft). We convert everything internally and show some alternative units.
Formula (LaTeX) + variables + units
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Area: \( A = 2(1 + \sqrt{2})s^2 \) Perimeter: \( P = 8s \) Inradius (apothem): \( r = \frac{s}{2 \tan(\pi/8)} \) Circumradius: \( R = \frac{s}{2 \sin(\pi/8)} \)
\( s = S(\sqrt{2} - 1) \) and so \( S = \dfrac{s}{\sqrt{2} - 1} \)
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.