Spiral Length Calculator (length of a roll, coil or Archimedean spiral)

Understanding the spiral length calculator — concept and principle

This calculator estimates how much material is wound on a roll, such as paper, tape, film, cable or fabric, from the core diameter, the outer diameter and the thickness of the material. It gives the length, the number of turns, a simple cross-check estimate and the spiral pitch. Use it for stock control, printing, converting, packaging and checking a supplier's roll length.

1,120 turns of about 590.6 mm each add up to about 661.5 m of material.

How it is calculated

The material forms an Archimedean spiral r = b·θ with b = t ÷ 2π.

Number of turns N = (D − d) ÷ 2t.
Exact length L = F(R) − F(r₀), where R = D/2, r₀ = d/2 and F(r) = [r·√(r² + b²) + b²·ln(r + √(r² + b²))] ÷ 2b.
Simple estimate: N × π(D + d)/2, the number of turns times the average circumference.

Diameters and thickness are entered in millimetres; the length is in metres.

The principle

Each turn adds one material thickness to the radius, so the radius grows evenly with the angle: that is an Archimedean spiral. Its length is the arc-length integral ∫√(r² + (dr/dθ)²) dθ, which has the closed form above. For thin material the radius changes very little in one turn, so each turn is almost a circle, and the total is close to "turns × average circumference". That is why both numbers agree here to the shown precision.

Worked example

Default inputs: core 76 mm, outer diameter 300 mm, thickness 0.1 mm.

Turns N = (300 − 76) ÷ (2 × 0.1) = 224 ÷ 0.2 = 1,120.
Average circumference = π × (300 + 76) ÷ 2 = π × 188 ≈ 590.62 mm.
Estimate: 1,120 × 590.62 mm ≈ 661,494 mm = 661.494 m.
Exact spiral integral with b = 0.1 ÷ 2π ≈ 0.0159155 mm per radian: 661.494 m.

Each point is the calculator re-run with only “Thickness of the material per turn (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Thickness of the material per turn (mm)	Length of material (m)
0.025	2,645.97
0.05	1,322.99
0.075	881.992
0.1	661.494
0.125	529.195
0.15	440.996
0.2	330.747

How to read the result

The length assumes the material is wound tightly with no gaps and a constant thickness. Real rolls are often wound slightly loosely or compress under tension, so measured lengths can differ by a few percent. If you know the true length, you can use the tool in reverse to estimate the effective thickness.

Common mistakes

Entering radius instead of diameter for the core or roll.
Using the nominal thickness of a material that compresses or has air between layers.
Mixing millimetres and inches.
Measuring the outer diameter of an oval or crushed roll.

Keep in mind: Results are estimates. For billing, contracts or production planning, confirm with the manufacturer's stated length or by weighing a known length of material.

Frequently asked questions

How do I calculate the length of paper on a roll?

Length ≈ π × (D² − d²) ÷ (4t), which is the same as turns times average circumference.

Why does the thickness matter so much?

Halving the thickness doubles the number of turns, and so doubles the length.

Can I find the thickness if I know the length?

Yes: t ≈ π(D² − d²) ÷ (4L), with all lengths in the same unit.

Sources: J. D. Cook, "Length of a general Archimedean spiral" (2024)

Checked 9 Nov 2026. J. D. Cook, “Length of a general Archimedean spiral” (2024) — spiral r = bθ and its length ≈ ½·bT² for large T; exact length from the polar arc-length integral ∫√(r² + (dr/dθ)²) dθ (standard formula).

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