Spiral Length Calculator — Archimedean Spiral Arc Length
Calculate the exact arc length of an Archimedean spiral (r = r₁ + bθ) from the inner radius, outer radius, and number of turns. Useful for clock springs, flat coils, spiral gaskets, heat-exchanger coil stock, and any equally-pitched flat spiral.
Exact arc length along the Archimedean spiral (same units as the radii)
- 1
Total angle Θ
2 × π × 5 = 31.415927 - 2
Radial growth per radian b
(10 − 0) ÷ 31.415927 = 0.31830989 - 3
Antiderivative F(r₂, b)
50.177102F(u) = u√(u²+b²)/2 + (b²/2)·ln(u + √(u²+b²)) - 4
Antiderivative F(r₁, b)
-0.057993 - 5
Arc length L
(50.177102 − -0.057993) ÷ 0.31830989 = 157.8182
How does this calculator work?
Arc length of an Archimedean spiral: set b = (r₂−r₁)/(2πn), then L = [F(r₂)−F(r₁)]/b where F(u) = u√(u²+b²)/2 + (b²/2)·ln(u+√(u²+b²)). Quick approximation: L ≈ π·n·(r₁+r₂). Exact for clock springs, flat coils, and spiral gaskets — all distances in the same unit.
Formula
How this is calculated
The Archimedean spiral is the curve r(θ) = r₁ + bθ in polar coordinates — the radius grows linearly with angle, so each successive turn is equally spaced. Given the inner radius r₁, outer radius r₂, and n turns, the radial growth rate is b = (r₂ − r₁)/(2πn). The arc length integral ∫₀^{2πn} √(r(θ)² + b²) dθ is evaluated in closed form by substituting u = r₁ + bθ, converting it to ∫_{r₁}^{r₂} √(u² + b²) du / b. The antiderivative is F(u) = u√(u²+b²)/2 + (b²/2)·ln(u + √(u²+b²)), giving L = [F(r₂) − F(r₁)] / b exactly.
The approximate formula L ≈ πn(r₁+r₂) treats each turn as a circle of the average radius. This is accurate to within a few percent when the pitch is small relative to the average radius, but the exact formula is always used for the primary result.
All inputs and outputs share the same length unit — the formula has no unit-specific constants. The XY plot shows how the radius grows with each turn, confirming the constant pitch.
Frequently asked questions
Measure the innermost winding radius (r₁) and outermost winding radius (r₂), and count the turns (n). Enter these values — the result is the total wire length. If r₁ is zero (wound tightly to a point), use a very small positive value like the wire radius.
The approximate formula πn(r₁+r₂) treats every turn as a circle of the mean radius. The error is proportional to (b/r̄)², where r̄ = (r₁+r₂)/2, so it is negligible for tightly wound coils (b ≪ r̄) but can reach a few percent for loosely wound ones.
No — this calculator handles a flat (2-D) Archimedean spiral where the radius grows and the height stays constant. For a 3-D helix of fixed radius r and pitch p per turn, use the simpler formula: total length = n × √((2πr)² + p²).
Also known as
TG we-Calculate Editorial Team. (2026). Spiral Length Calculator — Archimedean Spiral Arc Length [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/spiral-length-calculator
TG we-Calculate Editorial Team. "Spiral Length Calculator — Archimedean Spiral Arc Length." TG we-Calculate. 2026. https://we-calculate.com/calculator/spiral-length-calculator.
TG we-Calculate Editorial Team, "Spiral Length Calculator — Archimedean Spiral Arc Length," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/spiral-length-calculator
@misc{wecalculate_spiral_length_calculator, title = {Spiral Length Calculator — Archimedean Spiral Arc Length}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/spiral-length-calculator}}, year = {2026}, note = {TG we-Calculate} }
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