Helic Calculator
This page is a helix calculator: “helic” is the short slug, the geometry is a helix, the curve that winds around a cylinder while it climbs. Think springs, screw threads, augers, spiral staircases and DNA. Enter the radius or the diameter of the helix, the pitch (the height gained in one full turn) and the number of turns, and the tool returns the arc length, total height, circumference, helix angle, length per turn, unit rise, handrail radius, and two rough bounding-cylinder references. The length can be pinned down three ways: pitch with the number of turns, pitch with the total height, or turns with the total height, in which case the tool derives the pitch as height ÷ turns.
Geometric path length of the helix centerline. For wire coils, use the mean coil diameter (through the wire center), not the outside or inside diameter. Surface area and volume are rough bounding-cylinder references, not tube measurements.
Quick Answer
Arc length L = turns × √((2πR)² + pitch²), where R is the radius of the helix path. In the pre-filled example, radius 5 in, pitch 10 in and 3 turns give an arc length of 98.91 in, a total height of 30.00 in and a helix angle of 17.66°. Enter a diameter instead and the tool halves it first, so diameter 10 in returns the same 98.91 in. The same 98.91 in comes back from the two height entries: pitch 10 in with a total height of 30 in (turns derived as 30 ÷ 10 = 3), or 3 turns with a total height of 30 in (pitch derived as 30 ÷ 3 = 10 in).
Key takeaways
- One turn of a helix unwraps into a right triangle: circumference across, pitch up, length of the turn as the hypotenuse. Every number the tool prints comes from that triangle.
- Measure the path, not the part: for wire coils the radius is through the wire center (the mean coil size), not the outside or inside edge.
- The Units selector is a label, not a converter. The tool never converts between units, so type every measurement in the one unit you pick.
- Surface area and volume are rough bounding-cylinder references, not measurements of the tube or wire itself. The arc length is the number to cut or order by.
- Three entry modes pin down the same helix: pitch with turns, pitch with total height (turns = height ÷ pitch), or turns with total height (pitch = height ÷ turns).
- Quick Answer
- How to use the helix (helic) calculator
- The formulas behind the helix numbers
- Worked examples, verified against the tool
- Coils, springs, and the centerline rule
- Helic or helix length: which calculator do you need?
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the helix (helic) calculator
Set how the size is given, enter the size, pitch and turns in one unit, then click Calculate helix. Results stay hidden until you calculate, and Reset restores the worked example below.
- Size given as: choose Radius or Diameter. The label on the size field follows your choice, and a diameter is halved before any math runs.
- Size: the radius or diameter of the helix path itself. It must be greater than zero; blank, zero and negative entries stop with an error.
- Pitch: how far the helix climbs in one full turn, in the same unit as the size. Zero is allowed in Pitch and turns mode and means a flat circle; a negative pitch is rejected.
- Number of turns: decimals are fine, so 2.5 turns prices a half turn correctly. It must be greater than zero.
- Length entry: choose how the length is pinned down. Pitch and turns is the original entry. Pitch and total height derives the turns as height ÷ pitch, so a spec that lists only the finished height is enough; the pitch must be greater than zero in this mode, because the turns come from dividing by it. Turns and total height (derive pitch) derives the pitch as height ÷ turns and ignores the pitch field; the turns must be greater than zero, and a total height of zero derives a pitch of zero, the flat-circle case.
- Total height: the total axial height in the same unit as the size. It is read only in the two total-height entry modes and ignored in Pitch and turns mode.
- Units: millimeters, centimeters, meters, inches or feet. This only labels the results; it converts nothing.
- Read the badge: the result badge grades the helix angle as Shallow under 5°, Moderate from 5° to 20°, and Steep over 20°. Tight springs sit near 0°; augers and steep screws climb fast.
The formulas behind the helix numbers
Unwrap one turn of the helix and it becomes a right triangle: one leg is the circumference C = 2πR, the other is the pitch p, and the hypotenuse is the length of that turn. The same unwrapping gives the arc length of any smooth curve, the idea the general arc length calculator applies to other shapes.
Length per turn = √(C² + p²)
Arc length L = N × √(C² + p²), where N is the number of turns
Total height H = N × p
Turns from height N = H ÷ p (Pitch and total height mode)
Pitch from height p = H ÷ N (Turns and total height mode)
Helix angle α = atan(p ÷ C), measured from the plane perpendicular to the axis
Unit rise = p ÷ C
Handrail radius = (4π²R² + p²) ÷ (4π²R), the radius of curvature of one turn
Two outputs need honest labels. The volume is π × radius² × total height, the volume of the cylinder the helix fits inside. The surface figure is π × radius × arc length, carried over from the original tool as a rough bounding-cylinder reference; it is not the lateral surface of that cylinder (which would use the height) and not the surface of the wire or tube. Treat both as order-of-magnitude figures. The handrail radius is the helix's radius of curvature: it is always at least the radius itself, equals it when the pitch is zero, and grows as the pitch steepens.
Worked examples, verified against the tool
Every row below reproduces exactly in the calculator. The first row is the pre-filled example loaded when the page opens.
| Run | Inputs | Arc length | Height | Angle |
|---|---|---|---|---|
| Pre-filled example | Radius 5 in, pitch 10 in, 3 turns | 98.91 in | 30.00 in | 17.66° |
| Same helix by diameter | Diameter 10 in, pitch 10 in, 3 turns | 98.91 in | 30.00 in | 17.66° |
| Same helix by total height | Radius 5 in, pitch 10 in, height 30 in | 98.91 in | 30.00 in | 17.66° |
| Same helix, pitch derived | Radius 5 in, 3 turns, height 30 in | 98.91 in | 30.00 in | 17.66° |
| Wide, nearly flat coil | Radius 50, pitch 2, 1 turn | 314.17 | 2.00 | 0.36° |
| Flat circle (pitch zero) | Radius 5 cm, pitch 0, 3 turns | 94.25 cm | 0.00 cm | 0.00° |
The wide-coil row is a useful sanity check: circumference is 314.16 and the arc length is 314.17, because a 2-unit pitch barely lengthens a 314-unit loop. The full detail for the pre-filled run reads: circumference 31.42 in, length per turn 32.97 in, unit rise 0.31831, handrail radius 5.51 in, bounding-cylinder surface area (rough) 1,553.63 in² and volume (rough) 2,356.19 in³.
Coils, springs, and the centerline rule
If you are sizing wire for a coil spring, the radius to enter is the mean coil radius through the wire centerline, not the outside or inside diameter. Enter the outside diameter and every turn is priced too long; enter the inside and it comes up short.
A worked check: a spring wound on a 50 mm mean diameter with an 8 mm pitch and 10 turns. Choose Diameter, enter 50, pitch 8, turns 10, units mm. The tool returns an arc length of 1,572.83 mm (157.28 mm per turn, angle 2.92°), which is the wire the active coils need, before end coils. The same centerline logic runs through flat stock too: the coil length calculator unwinds rolled material from its diameters. Beyond springs, the same geometry sizes screw threads, auger flights, conveyor spirals and spiral-stair handrails, where the handrail radius output is the curvature a fabricator bends to.
Helic or helix length: which calculator do you need?
This page and the helix length calculator share the same core formula. This page now covers both input styles: it accepts the number of turns, the total height instead of the turns (deriving the turn count from height ÷ pitch), and turns with the total height (deriving the pitch from height ÷ turns), and it adds the full geometry on top: units label, helix angle, unit rise, handrail radius and the rough bounding-cylinder figures, with a pitch of zero accepted in Pitch and turns mode. The helix length page derives the turns from the height the same way, requires a positive pitch in every mode, and prints no handrail or cylinder figures. Neither tool converts units.
What competing calculators miss
The helix tools ranking for this keyword each stop partway, and two of them print formulas that are plainly wrong:
- Softonic's helix length calculator: the strongest rival, with radius or diameter, pitch, and turns or total rise, plus length per turn and angle. It stops there: no handrail (curvature) radius, no unit rise, and no worked detail beyond the headline length.
- BraveCalculator's helix and spiral length calculators: clean single-purpose tools that return the length of one turn and the full helix. Radius only, no diameter entry, no angle, no height readout next to the length.
- TrueGeometry's helix pages: one takes diameter, height and turns and derives pitch, which suits a different question. Its arc-length explainer prints L = π × r × n as the helix formula; that drops the pitch term entirely and even loses the circumference factor, so it understates every real helix. The tool above keeps the full √(C² + p²) term.
- CalculatorsConversion's helix length page: careful worked case studies for springs and staircases, but the calculator itself is thin and the numbers cannot be re-run with a diameter entry or a units label.
- The gap this page fills: radius or diameter entry, a units label, all eight outputs in one result (arc length, height, circumference, per-turn length, angle, unit rise, handrail radius, rough cylinder figures), turns-or-height entry with pitch derivation, a pitch of zero handled instead of rejected in turns mode, and every worked number on the page reproduced in the tool itself.
Frequently asked questions
L = N × √((2πR)² + p²), where R is the radius, p the pitch (rise per turn), and N the number of turns. Unwrap one turn into a right triangle: the circumference is one leg, the pitch is the other, and the length of one turn is the hypotenuse.
The axial distance the helix climbs in one complete revolution: total height divided by number of turns.
The angle between the helical path and the plane perpendicular to the helix axis: atan(pitch / circumference). A bigger pitch means a steeper angle.
Either; the tool converts. For a coil of wire, use the mean coil diameter measured through the wire center, then halve it for radius (or enter the diameter directly).
Pitch divided by circumference: the slope of the helix when one turn is unwrapped into a right triangle. Small values mean tightly wound coils; large values mean steep, screw-like paths.
Enter the mean coil radius, the pitch, and N. The arc length is the wire needed for the active coils; add extra for the end coils separately.
No. The selector is a label only: the tool computes with the numbers you typed and prints the chosen unit next to the results. It never converts between units, so enter the radius or diameter, the pitch and every other measurement in the one unit you selected.
Rough bounding-cylinder references carried over from the original tool, not measurements of the helical tube itself. The volume is π × radius² × total height; the surface figure is π × radius × arc length. Treat both as order-of-magnitude figures only.
At zero pitch the helix is a flat circle, so the arc length is the circumference times the turns and the total height is zero. Flat rings are the limiting case of a coil, so the tool accepts it instead of erroring. The exception is Pitch and total height mode, where the turns are derived as height divided by pitch, so a zero pitch there is rejected instead of dividing by zero.
Yes. Choose Pitch and total height under Length entry, enter the pitch and the total height, and the tool derives the turns as height divided by pitch before running the same formulas. A radius of 5 in with a 10 in pitch and a 30 in total height derives 3 turns and returns the same 98.91 in arc length as entering the turns directly. The pitch must be greater than zero in this mode.
Yes. Choose Turns and total height (derive pitch) under Length entry, enter the turns and the total height, and the tool derives the pitch as height divided by turns. Three turns over a 30 in total height derive a 10 in pitch and the same 98.91 in arc length at a 5 in radius. The turns must be greater than zero; a total height of zero derives a pitch of zero, the flat-circle case.
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References
- Weisstein, Eric W. Helix, from MathWorld, a Wolfram Resource. Accessed October 22, 2026.
- OpenStax, Rice University. Calculus Volume 3, Section 3.3: Arc Length and Curvature. Accessed October 22, 2026.
- Mathematics LibreTexts. 13.4: Arc Length and Curvature. Accessed October 22, 2026.
- Corral, Michael (Schoolcraft College). Vector Calculus, Section 1.9: Arc Length. Accessed October 22, 2026.
- OpenStax, Rice University. Calculus (OpenStax), Section 11.2: Calculus of Parametric Curves. Accessed October 22, 2026.