Spiral Length Calculator

Enter the inner radius, outer radius, and number of turns to compute the arc length of an Archimedean spiral using the exact closed-form formula.

Quick Facts

Spiral model
r = r₁ + bθ (Archimedean spiral)
Radius grows linearly with the angle swept.
Arc length formula
L = [r√(r²+b²) + b²ln(r+√(r²+b²))] / (2b)
Evaluated at the outer radius minus the inner radius.
Growth rate
b = (r₂ − r₁) / (2πn)
Coil spacing (pitch) equals 2πb per turn.

Your Results

Calculated
Spiral Length
-
Total arc length of the spiral
Coil Spacing (Pitch)
-
Radial distance between successive turns
Total Angle Swept
-
Across the number of turns entered
Growth Rate (b)
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Radius increase per radian

Ready

Enter the inner radius, outer radius, and number of turns, then press Calculate.

Formula and Method for Spiral Length

An Archimedean spiral is the curve traced when a point moves away from a center at a constant rate while it rotates at a constant angular speed — the kind of shape you see in a rolled-up tape measure, a coiled hose, or a groove on a vinyl record. In polar coordinates its radius grows linearly with the angle swept: r(θ) = r₁ + bθ, where r₁ is the starting (inner) radius and b is the growth rate, the amount the radius increases per radian. This calculator finds the exact arc length of that curve between an inner radius and an outer radius over a given number of turns.

How the calculation works

Arc length along any polar curve is found by integrating √(r² + (dr/dθ)²) dθ. For the Archimedean spiral, dr/dθ = b (a constant), so the integral becomes ∫√(r² + b²) dθ, which has a closed-form antiderivative. First the calculator finds the growth rate from your inputs: b = (r₂ − r₁) / (2πn), where n is the number of turns. Then it evaluates L = [r·√(r²+b²) + b²·ln(r + √(r²+b²))] / (2b) at the outer radius r₂ and subtracts the same expression evaluated at the inner radius r₁. If the outer and inner radius are equal (a pure circle repeated n times), the calculator instead uses the simpler L = n × 2π × r, since the general formula divides by a growth rate of zero in that case.

Common mistakes

  • Confusing radius with diameter: the inner and outer radius fields expect the distance from the center to the coil, not the full width across it — halve any diameter measurement first.
  • Mixing units: enter both radii in the same unit (millimeters, centimeters, meters, inches, or feet) before running the calculation.
  • Miscounting turns: a "turn" is one full 360° revolution; a half-wound coil is 0.5 turns, not 1.

Real-world applications

  • Estimating the length of tape, ribbon, or fabric wound onto a spool from its inner core radius, outer roll radius, and number of wraps.
  • Sizing coiled hoses, cables, or wire on a reel to know how much material is available before unwinding.
  • Approximating the groove length on a spiral-cut record or the wire length in a flat spiral spring.
  • Checking geometry homework or CAD sketches involving Archimedean spirals, scrolls, or volute shapes.

Frequently Asked Questions

What formula does this calculator use to find spiral length?
This calculator models an Archimedean spiral, where the radius grows linearly with the angle: r(θ) = r₁ + bθ. The exact arc length between the inner and outer radius comes from integrating √(r² + b²) dθ, which has the closed form L = [r·√(r²+b²) + b²·ln(r + √(r²+b²))] / (2b), evaluated at the outer radius minus the inner radius.
How is the growth rate (pitch) determined from the number of turns?
The growth rate b equals the increase in radius per radian: b = (r₂ − r₁) / (2π × n), where n is the number of turns. The spacing between two successive coils, called the pitch, equals 2πb.
What happens when the inner and outer radius are equal?
If the outer radius equals the inner radius, the path is not a growing spiral but a circle traced n times, so the length simplifies to L = n × 2π × r, the circumference multiplied by the number of turns.
Does this work for rolled material like tape, paper, or wire coils?
Yes. Material wound into evenly spaced concentric loops, such as a roll of tape or a coiled wire, closely approximates an Archimedean spiral, so this formula also gives an accurate estimate of the total length of rolled material.

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