Helix Length Calculator
This calculator assumes constant radius and constant axial pitch. It does not model a 2D spiral, tapered or elliptical helix, variable pitch, deformed coil or arbitrary spline.
Set the centerline size
Use the diameter or radius of the path followed by the material centerline, not automatically an outside or inside coil diameter.
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Calculate the centerline length of a constant-radius circular helix from its radius or diameter, axial pitch, and either number of turns or total rise. The calculator also reports length per turn, total axial rise, circumference, and helix angle. It accepts partial turns and keeps all linear dimensions in one selected unit. Use the radius or diameter of the path itself. For a coil made from round wire, that is normally the mean coil diameter through the wire centerline, not the outside or inside diameter. This is a geometric path-length calculation, not a spring, structural, or manufacturing design.
How to calculate circular helix length
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1
Choose the unit and centerline size
Select one shared length unit, then enter either the helix centerline diameter or radius. For round-wire coils, use the mean diameter measured through the wire centerline.
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2
Enter the axial pitch
Pitch is the distance the helix advances along its axis during one complete turn. It is not the sloping distance along the helix.
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3
Set turns or total rise
Enter the number of turns, including a decimal for a partial final turn, or enter total axial rise. With pitch fixed, the calculator derives the other value.
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4
Read and check the results
Use total centerline length as the main result, then review length per turn, total rise, circumference and helix angle. Confirm that the dimensions describe a constant-radius, constant-pitch path.
Circular helix length formula
A circular helix has a constant centerline radius and a constant axial rise per revolution. If the helix is unwrapped for one complete turn, its path is the hypotenuse of a right triangle. One leg is the circular distance 2πR, and the other is the axial pitch p.
Using centerline radius R, centerline diameter D = 2R, pitch per turn p, and number of turns N:
circumference C = 2πR = πDlength per turn ℓ = √(C² + p²)total helix length L = N × √((2πR)² + p²)total rise H = N × phelix angle α = atan(p ÷ C)
The angle is measured from the plane perpendicular to the helix axis. A larger pitch makes the path steeper. Right-handed and left-handed helices of the same radius, pitch magnitude and turns have the same length.
If total rise H is entered instead of turns, then:
N = H ÷ pL = √((2πRN)² + H²)
These are equivalent forms of the same formula. Keep R, D, p, H and L in compatible length units. The number of turns is dimensionless.
Worked example with three turns
Suppose a cylindrical helix has a centerline diameter of 10 mm, a pitch of 5 mm per turn, and 3 turns.
- Circumference:
C = π × 10 = 31.4159265 mm - Length per turn:
ℓ = √(31.4159265² + 5²) = 31.8113257 mm - Total length:
L = 3 × 31.8113257 = 95.4339770 mm - Total rise:
H = 3 × 5 = 15 mm - Helix angle:
α = atan(5 ÷ 31.4159265) ≈ 9.043061°
The total helix centerline length is about 95.434 mm. Notice that it is longer than three circumferences because each turn also rises by 5 mm.
Worked example with a partial turn
Now use a centerline diameter of 8 cm, a pitch of 2 cm, and 0.25 turn.
C = π × 8 = 25.1327412 cmℓ = √(25.1327412² + 2²) = 25.2121931 cmL = 0.25 × 25.2121931 = 6.3030483 cmH = 0.25 × 2 = 0.5 cm
The quarter-turn path is about 6.30305 cm long and rises 0.5 cm. Turns do not need to be whole numbers as long as pitch remains the rise for one full revolution.
Centerline diameter and spring coils
The formula follows the helix centerline. For a round-wire coil, use the mean coil diameter Dmean, which passes through the center of the wire:
Dmean = outside diameter − wire diameterDmean = inside diameter + wire diameter
Using the outside diameter directly makes the calculated body path too long; using the inside diameter makes it too short. The result covers only the regular helical body described by the inputs. A real spring or formed-wire part can also contain inactive end coils, hooks, loops, ground ends, transition bends, straight sections and manufacturing allowance. Add those from an appropriate drawing or qualified process specification. This calculator does not determine spring rate, load, stress, fatigue life, buckling, material suitability or an exact stock cut length.
A helix is not every curved or coiled path
| Shape | Defining geometry | Appropriate length method |
|---|---|---|
| Circular helix | Constant radius and constant axial pitch | L = N × √((2πR)² + p²) |
| Circular arc | Constant planar radius through angle θ |
`L = R × |
| 2D spiral | Radius changes as the angle changes | Polar arc-length integral ∫√(r² + (dr/dθ)²)dθ |
| Conical helix | Radius changes along the axis | Variable-radius curve calculation |
| Variable-pitch helix | Axial rise per turn changes | Parametric arc-length calculation |
A flat Archimedean spiral is not a circular helix viewed from above. Its radius grows with angle, so the constant-radius helix formula does not apply. In turns mode, a pitch of zero is accepted as a repeated-circle limiting case rather than a three-dimensional helix. Total-rise mode requires positive pitch and positive rise so the number of turns is defined.
Scope and limitations
The calculator assumes an ideal circular centerline with constant radius and constant pitch. It does not cover tapered or conical helices, elliptical helices, variable pitch, deformed coils or arbitrary spline approximations. Measurements must describe the same centerline and use compatible units. Surface paths around a thick cylinder also need a clearly defined reference radius.
For fabricated parts, stairs, handrails, augers, threads, springs, tubing and structural components, geometry is only one input to a safe design. Allowances, bend behavior, end details, tolerances, loads, material properties, connections and applicable standards require suitable engineering or fabrication guidance. Do not treat centerline length as approval of a design or as a guaranteed purchase quantity.
Official technical references
- OpenStax Calculus Volume 3: Arc Length and Curvature: derives the vector arc-length formula and applies it to a circular helix.
- OpenStax Calculus Volume 2: Arc Length in Polar Coordinates: gives the separate arc-length integral required for a planar polar curve such as a 2D spiral.
- Autodesk helix properties: defines helix radius, turns and turn height as the distance between turns.
- Autodesk overview of helices: distinguishes cylindrical, conical and flat helix geometry.
- NIST SI units for length: documents SI length units and the exact relationship
1 in = 25.4 mm.
Frequently Asked Questions
No. Pitch is the axial rise during one complete turn. Length per turn is the sloping centerline distance and equals the square root of the circumference squared plus the pitch squared.
Yes. A decimal number of turns describes a partial final revolution. For example, 2.25 turns means two complete turns plus one quarter-turn, and the total rise is 2.25 times the pitch.
Use the mean coil diameter through the wire centerline. For round wire, subtract one wire diameter from the outside diameter or add one wire diameter to the inside diameter. End features and manufacturing allowances are not included.
No. A flat spiral normally changes radius as its angle changes and needs a polar arc-length calculation. This tool assumes a constant-radius circular path. Turns mode accepts zero pitch only as the repeated-circle limiting case; total-rise mode requires positive pitch and rise.
No. Left-handed and right-handed helices have the same centerline length when their radius, pitch magnitude and turn count are equal. Direction matters for geometry and assembly, but not for this length formula.
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