Welcome to the spiral length calculator, a universal tool that lets you compute the length of an Archimedean spiral and its number of turnings from a few easily measured dimensions. In everyday life you can use it to find the length of anything that comes in a rolled-up form: rolls of paper, tape, film, fabric, cable, and so on. It also measures the length of a spiral wound helically around a cylinder.
What is a spiral and the Archimedean spiral?
The definition of a spiral tells us that it is the trajectory of a point moving around a center point while continuously moving away from (or towards) it. Many named spirals have been described — such as the Pythagorean or Fibonacci spiral — but the Archimedean spiral is a unique spiral named after the ancient Greek mathematician Archimedes.
Starting from a central point, you draw a curved pattern that gets progressively farther away as it revolves around that point. Archimedes described it in his work On Spirals. In polar coordinates it can be written as:
where:
- a — a parameter marking the initial radius of the spiral (the location of the center point);
- b — a constant controlling the distance between successive turns;
- r — the distance from the beginning of the spiral; and
- θ — the rotation angle from the start.
💡 The Archimedean spiral was one of only a few curves beyond the straight line and the conic sections that were known in antiquity. Although Archimedes did not discover the spiral, he described it to study the quadrature of the circle and the triangle, so the curve was named after him.
A spiral has some basic properties:
- Outer diameter D;
- Inner diameter d;
- Thickness t (the distance between arms, i.e. the separation distance);
- Length of the spiral L; and
- Number of turnings N.
Spiral length formula
To keep things simple, the roll method uses the approximate formula. Imagine the spiral as a series of concentric circles (one per turn), where the radius increases by the thickness of the tape with every turn. This simplified roll length formula works well when the thickness is small compared to the smaller diameter of the roll.
The number of turns N is just the difference between the two radii divided by the tape thickness t. In terms of diameters — the outer diameter D, inner diameter d, and thickness t — the number of turns is:
The total roll length follows from summing the circumferences of all the concentric circles:
What is the spiral length from the Archimedean spiral equation?
If you know the spiral equation r = a + bθ, the calculator uses the exact arc-length integral in polar coordinates:
The angle θ sweeps from 0 to 2πN, where N is the number of turns. The separation distance between successive arms equals t = 2πb. In the Equation (Archimedean) mode, enter the starting radius a, the growth b (radius gained per radian), and the number of turns N.
How to calculate the length of spiral around a cylinder
For any helically rolled project — a wire wrapped around a rod, a stripe on a barber pole, or a thread on a bolt — the spiral is a helix. If you "unroll" one full turn, it becomes the hypotenuse of a right triangle whose legs are the circumference πD and the pitch (rise per turn). The total length over N turns and height H is:
In the Cylinder (helix) mode, enter the cylinder diameter D, the number of turns N, and the winding height H.
How to use this spiral length calculator
- Choose a method — a rolled spiral from its diameters, the Archimedean spiral equation, or a helix around a cylinder.
- Pick your unit system — metric (cm) or US/Imperial (in).
- Enter the required dimensions for the method you selected.
- Read off the spiral length together with the number of turnings and related properties.
What will be the error if I use the exact formula in polar coordinates?
The concentric-circle (roll) formula is an approximation, while the polar-coordinate integral is exact. For real rolls the difference is tiny: when the tape thickness t is much smaller than the inner diameter d, the approximation and the exact result agree to a fraction of a percent. The error grows only when the roll has very few turns or an unusually thick tape. For most rolls of paper, tape, and film, the simple formula is more than accurate enough.
Unit Systems
Metric System
Enter the dimensions in centimeters (cm). The spiral length is also shown in millimeters (mm), meters (m), and inches (in).
US / Imperial System
Enter the dimensions in inches (in). The spiral length is also shown in feet (ft), yards (yd), and meters (m).
Example: length of a roll of tape
A roll of tape has an outer diameter D = 20 cm, an inner (core) diameter d = 10 cm, and a tape thickness t = 0.1 cm.
L = π × (20² − 10²) / (4 × 0.1) = π × 300 / 0.4 ≈ 2,356 cm ≈ 23.6 m
FAQs
What is the Archimedean spiral equation?
In polar coordinates it is r = a + bθ, where a is the starting radius, b controls how quickly the arms spread apart, and θ is the rotation angle. Successive arms are separated by a constant distance t = 2πb.
How do I find the number of turns of a roll?
Divide the difference between the outer and inner radii by the material thickness: N = (D − d) / (2t), using the outer diameter D, inner diameter d, and thickness t.
Can I compute the length of tape, paper, or film?
Yes. Any material wound into a roll forms an approximate Archimedean spiral, so the roll method gives its length directly from the outer diameter, the core (inner) diameter, and the material thickness.
Is the length of a spiral the same as a helix?
Not quite. A flat spiral (like a roll) lies in a plane, while a helix winds around a cylinder and also rises along its axis. Use the Cylinder (helix) method for anything wrapped around a rod, pipe, or bolt.