Spiral Length Calculator
Enter the outer diameter, inner diameter, and material thickness to find the total length and number of turns of a flat (Archimedean) spiral, such as a roll of tape, paper, or film. Switch to Helical mode to calculate the length of a cylindrical coil, spring, or staircase ramp. Results update as you type.
How the Archimedean spiral length formula works
An Archimedean spiral is a flat coil where each successive loop is spaced by a constant radial distance equal to the material thickness t. Given an outer diameter D, an inner diameter d, and thickness t, the number of turns is N = (D - d) / (2 x t), because each turn adds one thickness on each side of the winding. The average circumference of a single turn is pi x (D + d) / 2, the circumference at the mean diameter. Multiplying by N gives the total length: L = pi x N x (D + d) / 2. This approximation treats each turn as a perfect circle of the mean diameter and is accurate to within 0.1 % for tightly wound coils such as paper or tape rolls.
How the helical (3D) spiral length formula works
A helix wraps around the outside of a cylinder of diameter D and height H. When you unroll a single turn of the helix, it becomes the hypotenuse of a right triangle whose base is the circumference pi x D and whose height is the axial pitch (H / N). For N complete turns, the horizontal leg becomes N x pi x D, and the total length is L = sqrt(H^2 + (N x pi x D)^2). This is exact for a uniform cylindrical helix with constant pitch, such as a coil spring or the handrail of a spiral staircase.
Real-world uses of the spiral length calculator
Knowing how much material is on a roll is essential in manufacturing, packaging, printing, and construction. Tape, paper, film, cable, wire, and sheet-metal coil stock are all stored wound into Archimedean rolls. Measuring the outer diameter, inner core diameter, and layer thickness lets you calculate the remaining length without unwinding the roll. For 3D applications, the helical formula is used for designing coil springs with a target free length, calculating the railing length for a spiral staircase, estimating wire in a toroidal coil, or routing coolant tubing around a cylinder.
Accuracy and practical notes
The Archimedean formula assumes the material compresses uniformly and each layer has exactly the same thickness throughout the roll. In practice, wound materials like paper can compress slightly under tension, so the actual length may differ by 1 to 2 % from the calculated value for very thick stock. Measure the core diameter to the inside edge of the innermost winding, not the bare core, to get the correct d value. For the helix formula, the result is exact only when the helix is uniform, meaning constant diameter and constant pitch. Tapered coils (conical helices) require integration of the arc-length formula and cannot be solved with a simple closed form.
Typical spiral dimensions by application
| Application | Outer diam. | Inner diam. | Thickness | Approx. length |
|---|---|---|---|---|
| Masking tape (18 mm wide) | 60 mm | 40 mm | 0.13 mm | ~50 m |
| Adhesive tape roll | 90 mm | 40 mm | 0.05 mm | ~550 m |
| Newspaper roll | 400 mm | 76 mm | 0.10 mm | ~5 km |
| 35 mm film canister | 64 mm | 30 mm | 0.14 mm | ~10 m |
| Steel coil stock (3 mm) | 1500 mm | 508 mm | 3 mm | ~520 m |
Approximate ranges for common wound products. Use these as a sanity check for your inputs.
Frequently asked questions
What is the formula for the length of a spiral?
For a flat Archimedean spiral, the length is L = pi x N x (D + d) / 2, where N = (D - d) / (2 x t) is the number of turns, D is the outer diameter, d is the inner diameter, and t is the material thickness. For a cylindrical helix, the length is L = sqrt(H^2 + (N x pi x D)^2), where H is the cylinder height and D is its diameter.
How do I measure the thickness of a very thin material like tape?
Stack 10 or 20 layers of the material, measure the total thickness with a calliper or a ruler, and divide by the number of layers. For typical household tape, this gives about 0.05-0.15 mm per layer. An alternative is to measure a roll of known length, then back-calculate thickness using t = (D - d) / (2 x N).
Why does the formula use the mean diameter?
The Archimedean spiral expands linearly with each turn. The outermost turn has circumference pi x D and the innermost has pi x d. Because the circumference grows linearly from pi x d to pi x D over N turns, the total length equals the average circumference pi x (D + d) / 2 multiplied by the number of turns, N. This is the exact result of integrating the circumferences and not just an approximation of the mean.
Can I use this calculator for a coil spring?
Yes, use helical mode. Enter the outer diameter of the spring wire coil (not the wire diameter), the total free length of the spring as the cylinder height, and the number of active turns. The result is the total wire length needed to produce the spring. Remember to add two end-turn lengths if the spring has closed or ground ends.
How do I calculate the remaining material on a partial roll?
Measure the current outer diameter of the partial roll (this is your new D), keep the inner diameter d and thickness t the same, and recompute. The difference between the original length and the new length is the amount already used.
What is the difference between an Archimedean and a logarithmic spiral?
In an Archimedean spiral, the distance between successive turns is constant, so d = k x theta in polar form. This describes wound materials where each layer has the same thickness. In a logarithmic spiral, the gap between turns grows proportionally to the radius (r = a x e^(b x theta)), giving it the same shape at any scale. Seashells and galaxies follow logarithmic spirals. For practical wound-material calculations, the Archimedean model is almost always the correct one.