How to use the helix length calculator
- Enter the diameter of the coil, measured to the center of the wire. For a coil that changes size, also enter the diameter at the end.
- Enter the pitch, the rise along the axis for each full turn.
- Choose whether you know the number of turns or the total height, and enter it.
- Read the length along the helix, the turns, the height and the pitch angle. All values use the unit you chose for the inputs.
Formula
One turn of a straight helix: sqrt((pi x D)^2 + p^2), with D the diameter and p the pitch
Length = turns x sqrt((pi D)^2 + p^2); turns = height / pitch
Tapered helix: the radius changes linearly, r = r0 + k t, and z = c t with c = p / (2 pi)
Length = integral of sqrt(r^2 + k^2 + c^2) dt over t = 0 to 2 pi x turns, solved in closed form with asinh
Unroll one turn of a straight coil and it becomes a right triangle with legs pi x D and p, which is where the first line comes from.
A spring, a stair and a cone
A coil with a diameter of 20, a pitch of 5 and 10 turns has a height of 50. Each turn is sqrt((pi x 20)^2 + 5^2) = 63.0305, and the whole coil is 630.30483 long, with a pitch angle of 4.5499 degrees. A spiral stair with a 1800 mm diameter and a pitch of 600 mm that rises 3600 mm makes 6 turns, and its centerline is 34,119.652 mm long, or about 34.1 m.
A tapered coil that shrinks from a diameter of 40 to 10 over 8 turns with a pitch of 4 is 629.46485 long, which is a little more than the 629.1 of a straight coil of the average diameter 25 with the same pitch and turns, because the length of one turn is a curved function of its diameter, so narrow and wide turns together come to slightly more than average turns.
Using the length
The length along the helix tells you how much wire to cut for a spring or a coil, how much rail or tubing a spiral staircase needs, or how much cable wraps around a drum. Add a little for the ends and for how the material is formed. Because the measure is to the center of the wire, use the diameter of the path, which is the outside diameter minus one wire thickness.
For the stiffness of a coil spring, use the compression spring rate calculator. For the diagonal of a flat shape, use the Pythagorean theorem calculator, which is the same triangle that unrolls from one turn.
Straight and tapered coils
When the end diameter is empty or equal to the start diameter, the coil is a straight cylinder and the formula is exact and simple. When it differs, the coil is a cone and the radius changes smoothly with the angle. The page uses a closed form for the integral, so the result is exact apart from rounding, not an approximation by short segments.
The pitch angle shown is the angle between the wire and a flat circle around the axis at the average diameter. A small angle means the coil is nearly flat like a spiral, and a large one means it is steep like a screw thread. For other shapes with a regular outline, the regular polygon calculator finds sides and area.
Assumptions and limits
- The pitch is constant along the coil, and turns can be fractional.
- The diameter is the diameter of the path of the center line of the wire.
- The units are whatever you enter; all inputs must use the same unit.
- A tapered coil changes its radius at a constant rate per turn.
Frequently asked questions
How do I find the length of wire in a coil?
Multiply the number of turns by sqrt((pi x D)^2 + p^2), where D is the diameter and p the pitch. The page does this for you.
What is the pitch?
The distance the coil moves along its axis for one full turn.
Can the number of turns be a fraction?
Yes. Enter 2.5 for two and a half turns.
What if I know the height and not the turns?
Choose Total height. The turns are the height divided by the pitch.