How much paper is left on a roll, how long is a coiled hose or a spring wire? A rolled material forms an Archimedean spiral, and its length follows from the number of turns and the average circumference.
How to use the Spiral Length Calculator
- Shape
- Outer diameter D
- Inner diameter d
- Material thickness t
- Height H
- Number of turns N
Results update as you type, and everything is calculated in your browser.
How it works
For a roll, the number of turns is the radial build-up divided by the thickness, and the length is turns × average circumference. For a helix, each turn is the hypotenuse of a triangle with the circumference and pitch as legs.
Inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose one: 2D spiral (roll of material), 3D helix (coil, spring, staircase) |
| Outer diameter D | Units: cm, m, mm, in, ft, yd … |
| Inner diameter d | Units: cm, m, mm, in, ft, yd … |
| Material thickness t | Units: cm, m, mm, in, ft, yd … |
| Height H | Units: cm, m, mm, in, ft, yd … |
| Number of turns N | A number |
Example values
How the main result changes when Outer diameter D changes and everything else stays at the example values above:
| Outer diameter D (cm) | Length (cm) |
|---|---|
| 10 | 282.7433 |
| 15 | 1,264.491 |
| 20 | 2,638.9378 |
| 25 | 4,406.0837 |
| 30 | 6,565.9286 |
| 40 | 12,063.7158 |
Formula
Roll: N = (D − d)/(2t), L = π·N·(D + d)/2
Helix: L = N·√((πD)² + p²), p = H/N
Worked example
Inputs — Shape: 2D spiral (roll of material); Outer diameter D: 20 cm; Inner diameter d: 8 cm; Material thickness t: 0.1 cm.
Results — Number of turns = 60; Length = 2,638.937829 cm.
Key terms
- Archimedean spiral
- r = a + bθ: radius grows evenly with angle.
- Helix
- A 3D spiral of constant radius and pitch.
Frequently asked questions
How do I estimate the paper left on a roll?
Measure outer and core diameters and the sheet thickness, then use L = πN(D + d)/2.
Is a roll exactly an Archimedean spiral?
Very nearly – the radius grows by the thickness every turn.
How long is the wire in a spring?
N·√((πD)² + p²), using the mean coil diameter.
What is the pitch of a helix?
The height gained per turn.