# Spiral Length Calculator

URL: https://speedycalculator.com/calculators/math/spiral-length/
Last updated: 2026-10-30

> The spiral length calculator finds the length of material in a roll (an Archimedean spiral – tape, paper, cable, carpet) from its outer and inner diameters and thickness, or the length of a 3D helix such as a coil spring or spiral staircase.

## Inputs

- Shape
- Outer diameter D (cm, m, mm, in, ft, yd, km, mi)
- Inner diameter d (cm, m, mm, in, ft, yd, km, mi)
- Material thickness t (cm, m, mm, in, ft, yd, km, mi)
- Height H (cm, m, mm, in, ft, yd, km, mi)
- Number of turns N

## Results

- Number of turns
- Length
- Pitch (rise per turn)

## Formula

```
Roll: N = (D − d)/(2t),  L = π·N·(D + d)/2
Helix: L = N·√((πD)² + p²),  p = H/N
```

## Explanation

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

FieldWhat to enterShapeChoose one: 2D spiral (roll of material), 3D helix (coil, spring, staircase)Outer diameter DUnits: cm, m, mm, in, ft, yd …Inner diameter dUnits: cm, m, mm, in, ft, yd …Material thickness tUnits: cm, m, mm, in, ft, yd …Height HUnits: cm, m, mm, in, ft, yd …Number of turns NA 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)10282.7433151,264.491202,638.9378254,406.0837306,565.92864012,063.7158

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Archimedes' spiral](https://mathworld.wolfram.com/ArchimedesSpiral.html)
- [Wolfram MathWorld – Helix](https://mathworld.wolfram.com/Helix.html)