Octagon Calculator — Area, Perimeter & Diagonals
Find every dimension of a regular octagon — area, perimeter, circumscribed circle radius, inscribed circle radius (apothem), and all three diagonal lengths — from the side length, circumradius, or apothem.
Known measurement
Area = 2(1 + √2) × s² ≈ 4.8284 × s²
- 1
Side length s
5 - 2
s²
5 × 5 = 25 - 3
8 × s²
8 × 25 = 200 - 4
Area = 8s² ÷ (4 × tan 22.5°)
200 ÷ 1.656854 = 120.7107tan 22.5° is exact: √2 − 1, giving Area = 2(1 + √2) s².
How does this calculator work?
Regular octagon: Area = 2(1 + √2) s² ≈ 4.828 s²; Perimeter = 8s; Circumradius R ≈ 1.307 s; Inradius r ≈ 1.207 s; 20 diagonals in 3 distinct lengths (≈ 1.848 s, 2.414 s, 2.613 s); interior angles = 135°. Enter any one measurement to get all others.
Formula
How this is calculated
A regular octagon has 8 sides of equal length s and 8 equal interior angles of 135°. All its properties follow from the side length using trigonometry on the central triangle formed by two radii and one side. That triangle has a central angle of 360° / 8 = 45°, and half of it gives a right triangle with one angle of 22.5° (π/8).
The circumradius R (centre to vertex) is R = s / (2 sin 22.5°) ≈ 1.3066 s, and the inradius (apothem) r — the distance from centre to midpoint of a side — is r = s / (2 tan 22.5°) = s(1 + √2)/2 ≈ 1.2071 s. The area equals half the perimeter times the apothem: Area = (8s × r) / 2, which simplifies to 2(1 + √2)s².
A regular octagon has n(n − 3) / 2 = 20 diagonals in three distinct lengths: short diagonals (skip 1 vertex) of length ≈ 1.8478 s, medium diagonals (skip 2 vertices) of length (1 + √2) s ≈ 2.4142 s, and the long diameter diagonals (connecting opposite vertices) of length 2R ≈ 2.6131 s. Enter any one measurement — side length, circumradius, or inradius — and every other dimension is derived in the same unit.
Frequently asked questions
Each interior angle is (8 − 2) × 180° / 8 = 135°. The exterior angle is 45°. This is why regular octagons tile neatly with squares — the 135° interior angle and the square's 90° angle sum to 360° / 2 when they share an edge corner.
A regular octagon has n(n − 3) / 2 = 8 × 5 / 2 = 20 diagonals. They come in three distinct lengths: 8 short diagonals (≈ 1.848 s), 8 medium diagonals (≈ 2.414 s), and 4 long diagonals (≈ 2.613 s) connecting opposite vertices.
Area = 2(1 + √2) s² ≈ 4.8284 s², where s is the side length. Equivalently, Area = (8s × r) / 2 where r is the apothem (inradius). Using the general regular polygon formula: Area = (n s²) / (4 tan(π/n)) with n = 8 gives the same result.
Also known as
TG we-Calculate Editorial Team. (2026). Octagon Calculator — Area, Perimeter & Diagonals [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/octagon-calculator
TG we-Calculate Editorial Team. "Octagon Calculator — Area, Perimeter & Diagonals." TG we-Calculate. 2026. https://we-calculate.com/calculator/octagon-calculator.
TG we-Calculate Editorial Team, "Octagon Calculator — Area, Perimeter & Diagonals," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/octagon-calculator
@misc{wecalculate_octagon_calculator, title = {Octagon Calculator — Area, Perimeter & Diagonals}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/octagon-calculator}}, year = {2026}, note = {TG we-Calculate} }
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