There is no marking on the side of a roll telling you how much is left. What you can measure is the core, the outside, and the thickness of the material, and those three numbers are enough to recover the length to within a fraction of a per cent. This spiral length calculator does that, and it also runs the question backwards: given a length and a core, how big will the finished roll be?
It applies to anything wound in a flat spiral — film, foil, paper, tape, wire on a flat reel, sheet steel in a coil, cable on a drum, filament on a spool, weed fabric, underlay, roofing membrane. Arb Digital publishes it alongside a wide library of free geometry and materials calculators, because "how much is left on that roll" is a question that comes up on almost every site and in almost every workshop.
What This Spiral Length Calculator Does
In its default mode it takes the inner diameter of the core, the outer diameter of the roll and the thickness of one wrap, and returns the total length of material. Alongside it you get the number of turns, the mean diameter of the wrap, the cross-sectional area of material in the annulus, and an exact spiral arc length computed from the true Archimedean geometry rather than the approximation.
The second mode reverses the calculation. Enter a core diameter, a thickness and a length, and it returns the outer diameter that length will build up to. That is the useful direction when you are specifying a spool, checking whether a coil will clear a guard, or working out whether a replacement roll will physically fit the machine it feeds.
A packing efficiency field handles the real-world case where the wraps are not perfectly tight. At ninety per cent, the same outside diameter holds ten per cent less material, which is closer to how a hand-wound roll actually behaves than the ideal geometry.
How to Use It
- Measure the core. The inner diameter is the tube or drum the material is wound on, measured across, not around.
- Measure the outside of the roll. Take it in two directions and average if the roll is not perfectly round, which most part-used rolls are not.
- Get the thickness right. Measure a stack of ten or twenty layers with a caliper and divide. A single-sheet measurement on thin material is dominated by the instrument's own error.
- Set the packing efficiency if the roll is loosely wound. Leave it at 100 per cent for a machine-wound roll.
- Switch modes to solve for the outer diameter when you know the length instead — for example when specifying a new spool.
The Formula and How It Is Calculated
The elegant way to think about a roll is as a volume problem in two dimensions. Unroll it and the material becomes a long thin rectangle of length L and thickness t, with a cross-sectional area of L × t. Roll it back up and that same area becomes an annulus between the core and the outside, with area (π ÷ 4) × (D² − d²). Setting the two equal gives the whole calculation:
L = π × (D² − d²) ÷ (4 × t)
The number of turns is (D − d) ÷ (2t), because each wrap adds one thickness to the radius and therefore two to the diameter. The mean diameter is simply (D + d) ÷ 2, and multiplying π × mean diameter × turns gives exactly the same length — the two routes are algebraically identical, which is a satisfying check.
The exact figure comes from treating the wrap as an Archimedean spiral, the curve whose radius grows linearly with angle, described at Wolfram MathWorld's entry on the Archimedean spiral. Writing r = r₀ + aθ with a = t ÷ 2π, the arc length integral ∫√(r² + a²) dθ has the closed form (1 ÷ 2a) × [u√(u² + a²) + a² ln(u + √(u² + a²))] evaluated between the inner and outer radii. That integral form and the inverse hyperbolic functions behind it are catalogued in the NIST Digital Library of Mathematical Functions.
Worked example: a roll with a 3 inch core, a 12 inch outside diameter and material 0.005 inches thick. The annulus area is (π ÷ 4) × (144 − 9) = 106.03 square inches, so the length is 106.03 ÷ 0.005 = 21,206 inches, or 1,767 feet. The turn count is (12 − 3) ÷ 0.01 = 900, and the mean diameter is 7.5 inches, so π × 7.5 × 900 = 21,206 inches as well. The exact spiral arc length agrees to better than one part in ten thousand.
Why the Approximation Is Almost Always Good Enough
The mean-diameter method treats every wrap as a perfect circle at its own radius, ignoring the small radial step where the spiral advances. The exact arc length accounts for that step. The gap between them is governed by the ratio of the thickness to the radius, and on any real roll that ratio is tiny — on the example above, the material is one three-thousandth of the core radius.
Where it starts to matter is at the other extreme: very thick material on a very small core, like a few turns of heavy hose on a narrow reel. With four turns rather than nine hundred, the spiral is visibly not a set of concentric circles and the exact figure earns its keep. Both numbers are shown so you can see when they diverge, which is more informative than picking one and hiding the other.
Where the Error Actually Comes From
Not from the formula. In practice the dominant error is the material thickness, because the length divides by it: a five per cent error in caliper is a five per cent error in the answer, and on thin film a five per cent error is easy to make with a tape measure and hopeful arithmetic. The second largest is the roll not being wound tight, which is exactly what the packing efficiency field exists to absorb.
Diameter errors matter much less than people expect, because the formula uses the difference of squares and both diameters are measured with the same instrument, so the errors partly cancel. If you have one measurement to take carefully, take the thickness — and take it across many layers rather than one.
How This Differs From an Arc Length Calculation
It is worth being precise about the boundary, because the two problems look similar and are not. Our arc length calculator measures the distance along part of a circle: the radius is constant, and the answer is radius × angle in radians. A spiral has no constant radius — that is its defining property — so the arc length formula does not apply, and using it with a mean radius gives the approximation on this page rather than the exact answer.
Related but different again: the circle calculator handles circumference, area and radius relationships for a single circle, and the hollow cylinder calculator gives the volume and wall properties of the annulus a roll occupies without saying anything about how long the wound material is. A three-dimensional helix, where the curve also advances along an axis, is a different shape again — the thread form on a fastener is the everyday example, covered by our thread pitch calculator.
Practical Uses for the Reverse Mode
Solving for the outer diameter answers the questions that arise before you buy rather than after. Will a 500 metre reel of this cable fit the drum? How large does the take-up spool need to be at the end of the run? If we specify a larger core to reduce curl, how much bigger does the finished roll become for the same footage?
That last one is worth a moment. Increasing the core diameter while keeping the length fixed grows the outside diameter, but by less than you might guess, because the material is added to a larger radius where each turn holds more. Doubling the core on the worked example above lifts the outside diameter by well under double. Run both numbers before assuming a bigger core will not fit. The cylinder volume calculator is the tool if you need the volume of the resulting roll rather than its diameter.
Arb Digital publishes free calculators for geometry, quantities and workshop maths. Browse the library, or tell us what is missing.
Browse the free tools hub Contact Arb DigitalCommon Mistakes to Avoid
- Entering radii instead of diameters. Both fields want the full width across the roll. Using radii understates the length by a factor of four.
- Measuring a single layer with a tape. On thin material the caliper of one sheet is below the resolution of the instrument. Measure many and divide.
- Forgetting the coating or the liner. If a release liner goes round with the material, its thickness is part of the wrap and belongs in the figure.
- Mixing units between fields. Diameters, thickness and the known length must all be in the same unit, and the tool converts only at the point of reporting.
- Assuming a part-used roll is still round. Rolls go oval in storage. Measure in two directions and average before you calculate.
Related Free Tools From Arb Digital
For distance along a fixed-radius curve use the arc length calculator, for single-circle relationships the circle calculator, for the annulus itself the hollow cylinder calculator or the cylinder volume calculator, and for the helical case the thread pitch calculator. Everything else is in the free tools hub.
Frequently Asked Questions
Measure the outer diameter, the core diameter and the thickness of one layer, then use L = π × (D² − d²) ÷ (4t). A 12 inch roll on a 3 inch core in 0.005 inch material holds about 21,206 inches, or 1,767 feet.
Because the material occupies an annulus, and the area of an annulus is the outer circle minus the inner one. Unrolled, that same area is a long rectangle of length times thickness, so dividing the annulus area by the thickness gives the length.
The mean-diameter result is an excellent approximation that treats each wrap as a circle. The page also computes the exact Archimedean spiral arc length, and on any normal roll the two agree to a small fraction of one per cent. They diverge only when the material is very thick relative to the core.
The material thickness, because the length divides by it — a five per cent error in caliper is a five per cent error in the result. Measure a stack of ten or twenty layers and divide rather than trusting a single-sheet measurement.
The outer diameter minus the inner diameter, divided by twice the thickness, since each wrap adds one thickness to the radius and therefore two to the diameter. The example above works out at 900 turns.
Yes, treating the cable diameter as the layer thickness, but expect a loose fit to the ideal because round cable does not stack like flat film. Use the packing efficiency field to allow for the air between the turns.
Not directly. A flat spiral stays in one plane while its radius grows; a helix keeps a constant radius while advancing along an axis. They are different curves with different length formulas, and the thread pitch calculator covers the helical case.
Results are geometric estimates produced from the dimensions you enter. Real rolls vary in tension, layer thickness and roundness, so treat the figure as an estimate rather than a measured quantity where the exact length matters commercially.
