Spiral Length Calculator
Calculate the arc length of an Archimedean spiral, a rolled strip, or a 3D helix from radii, turns, pitch, diameter, thickness, or the equation r = a + b theta.
Spiral length, roll length, and helix length
A spiral length calculator estimates the distance along a spiral curve. This page focuses on the Archimedean spiral, where radius changes at a constant rate as the angle increases.
A spiral length calculator finds the total length of a spiral from its radius, turns, and spacing. For an Archimedean spiral, use r = a + b theta and calculate length with an integral over theta. Spiral length depends on spiral type, starting radius, ending radius, and number of rotations.
Use it for spiral layouts, coils, scroll patterns, tubing, garden paths, cable wraps, flat springs, craft templates, and geometry problems where the spiral can be modeled as r = a + b theta.
The calculator supports five practical inputs: inner and outer radius with turns, start radius with pitch per turn, direct polar equation values, rolled material from inner diameter and thickness, and 3D helix length around a cylinder.
Spiral length result
--
Spiral length
--
--
Radii
--
--
Turns and angle
--
--
Pitch and average radius
--
--
Step-by-step work
--
--
--
Note: Flat modes model an Archimedean spiral with constant radial spacing. Helix mode assumes a constant cylinder radius and steady vertical rise. Real material length can differ if thickness, bends, overlap, manufacturing tolerance, or a non-Archimedean spiral shape is involved.
How to use the spiral length calculator
- Choose an input method: Use radii and turns, pitch and turns, direct equation values, rolled material dimensions, or 3D helix dimensions.
- Enter consistent units: Use the same unit for every radius and pitch input, such as inches, millimeters, or meters.
- Calculate: The result shows arc length, start and end radius, turns, angle swept, pitch, and average radius.
- Check the preview: The canvas drawing gives a quick visual check of the number of turns and whether the spiral expands outward or inward.
- Add material allowance: For real coils, tubing, or paths, add extra length for trimming, connectors, bends, or installation margin.
Spiral length formula
For an Archimedean spiral, radius changes linearly with angle: r = a + b theta. The arc length comes from the polar arc length formula.
r = a + b theta
Length = integral sqrt(r^2 + (dr/dtheta)^2) dtheta
pitch = 2pi b
When b is zero, the spiral is a circular arc and the length is simply radius x angle in radians. When b is not zero, the calculator uses the exact closed-form integral for r = a + b theta.
Polar arc length reference: Mathematics LibreTexts - Area and Arc Length in Polar Coordinates.
Exact length vs quick spiral estimates
| Method | What it assumes | Best use | Why it matters |
|---|---|---|---|
| Exact polar integral | The curve follows r = a + b theta | Math, CAD checks, precise flat spiral layouts | It accounts for both radial change and angular travel along the curve. |
| Average-radius estimate | Length is close to circumference at average radius times turns | Fast mental checks and rough material budgets | It is simple, but can drift when the spiral has few turns or large spacing. |
| Unwrapped helix | A cylinder unwraps into a rectangle and the helix becomes a diagonal | 3D spiral wraps, rails, hoses, coils around posts | It includes vertical rise, which a flat spiral formula cannot capture. |
For ordering material, use the exact result as the baseline and then add project-specific allowance for waste, connectors, bend radius, or cutting error.
Which input method should you use?
| Input method | Use it when you know | Best for | Watch for |
|---|---|---|---|
| Inner + outer radius | Start radius, end radius, and total turns | Flat coils, garden paths, circular layouts | Outer radius must differ from inner radius unless pitch is zero. |
| Start radius + pitch | Radial spacing between consecutive turns | Coils, springs, cable wraps, tubing spacing | Negative pitch creates an inward spiral. |
| Equation | a, b, start theta, and end theta | Math and engineering problems | Theta must be entered in radians. |
| Inner diameter + thickness | Core diameter, material thickness, and wraps | Tape, strip stock, gasket material, paper, film, flat roll estimates | Thickness is treated as the radial build per turn. |
| 3D helix | Cylinder radius, total height, and turns | Coils around posts, helical wraps, spiral rails, wrapped hose paths | The preview is a top-down projection, not a 3D rendering. |
Archimedean spiral definition: Merriam-Webster - Archimedean spiral.
Pick the right spiral model before trusting the result
A single "spiral length" phrase can describe different geometry. This calculator separates the common cases so the input matches the physical object: a flat Archimedean spiral, a rolled strip, or a 3D helix around a cylinder.
Flat Archimedean spiral
Use this for a curve drawn on a plane where radius increases evenly with every turn. The calculator uses the exact polar arc length integral, which is stronger than a simple average-radius estimate.
Rolled material
Use this when you know the inner diameter, material thickness, and number of wraps. It converts diameter and thickness into an Archimedean centerline so you do not have to manually convert to radius and pitch.
3D helix
Use this for a path that climbs as it wraps around a cylinder. One turn becomes the hypotenuse of a right triangle made from circumference and vertical rise.
Material length checklist
The mathematical curve length is a clean baseline. Real-world projects often need an added allowance, especially when the spiral is made from wire, pipe, rope, edging, tubing, or flexible strip material.
End connections
Add extra length for fittings, terminals, solder points, overlap, anchors, or knots.
Material thickness
Measure along the centerline of the material. Inner and outer edges have different lengths when the material has thickness.
Waste allowance
For cutting or installation, add a small percentage so the final piece is not short after trimming.
Worked example: spiral from 2 to 10 units over 4 turns
Suppose a flat spiral starts at radius 2 units, ends at radius 10 units, and makes 4 full turns. The radius increases by 8 units over 4 turns, so the pitch is 2 units per turn.
Step 1
Convert turns to radians
4 turns = 8pi radians, or about 25.133 radians.
Step 2
Find b
b = radial change / angle = 8 / 25.133, or about 0.318 per radian.
Step 3
Apply arc length
Use the polar arc length integral for r = 2 + 0.318theta to get the curve length.
This example is useful for checking whether your input method is set up correctly: inner radius 2, outer radius 10, and 4 turns should match start radius 2, pitch 2, and 4 turns.
Spiral type comparison
| Spiral type | Radius behavior | Use this calculator? | Common examples |
|---|---|---|---|
| Archimedean spiral | Radius changes by a constant amount per turn | Yes | Flat coils, scroll templates, garden spirals, spiral paths |
| Circular arc | Radius stays constant | Yes, with pitch = 0 | Partial circles, fixed-radius bends |
| 3D helix | Radius stays constant while height changes | Yes, with 3D helix mode | Cylinder wraps, helical rails, coils around posts |
| Logarithmic spiral | Radius grows by a constant ratio | No | Growth patterns, scaling curves, some natural forms |
| Hand-drawn or irregular spiral | Spacing changes unpredictably | No | Artwork, freehand sketches, irregular manufactured parts |
Measurement conversions before entering values
Most spiral length mistakes come from mixing diameter with radius, using degrees instead of radians in equation mode, or measuring spacing edge-to-edge instead of centerline-to-centerline.
Diameter to radius
If your drawing gives diameter, divide by 2 before entering inner or outer radius.
Degrees to radians
For equation mode, convert degrees to radians with radians = degrees x pi / 180.
Spacing to pitch
Use centerline spacing between neighboring loops as pitch. Edge spacing may need thickness added first.
Polar coordinate reference: OpenStax Calculus Volume 2 - Polar Coordinates.
Interesting fact
In an Archimedean spiral, equal angle increases create equal radial spacing. That makes it different from a logarithmic spiral, where the spacing grows by a constant ratio instead. This is why Archimedean spirals are common in mechanical layouts and flat coils, while logarithmic spirals often appear in growth patterns and scaling designs.
Frequently Asked Questions
What is a spiral length calculator for arc length?
A spiral length calculator finds the arc length along a spiral curve. This calculator models an Archimedean spiral, where the radius changes by a constant amount for each full turn. It is useful for estimating the length of a coil, wire, tubing, hose, spring, or geometry curve when the spiral spacing is consistent.
What is pitch or spacing in a spiral coil?
Pitch is the radial spacing gained after one full revolution of the spiral. In the equation r = a + b theta, pitch = 2pi b because one full turn is 2pi radians. For a flat coil, pitch describes the distance between neighboring loops of wire, tubing, hose, or spring material.
Can this calculator handle inward spirals with decreasing radius?
Yes. If the end radius is smaller than the start radius, or if pitch is negative, the spiral moves inward. The arc length remains positive because length measures distance traveled along the curve. If you measured diameter instead of radius, divide the diameter by 2 before entering the start or end radius.
Is the spiral length formula exact for every spiral shape?
No. The formula used here is exact for the Archimedean equation r = a + b theta. Logarithmic spirals, clothoids, hand-drawn spirals, and manufactured parts with changing spacing need a different calculator, equation, or direct measurement. Always match the spiral type to the formula before using the result for material planning.
How much extra material should I add for wire, tubing, or hose?
That depends on the project and material. For wire, tubing, hose, or a spring, add extra length for trimming, fittings, end connections, bends, overlap, and installation errors. The calculator gives the mathematical centerline length of the spiral curve, not a project-specific waste allowance.
Can this calculate roll length from diameter and thickness?
Yes. Choose the inner diameter plus thickness mode when you know the core diameter, material thickness, and number of wraps. The calculator converts diameter to radius, treats thickness as the radial pitch per turn, and then uses the same spiral arc length calculation to estimate roll length. This is useful for tape, film, gasket material, paper, and other flat strip materials.
What is the difference between spiral length and helix length?
A flat spiral lies in a plane, so its length depends on radius, angle, and spacing. A helix is a 3D curve that wraps around a cylinder while moving upward or downward, so its length also depends on height. In helix mode, the calculator unwraps the cylinder: each turn becomes a right triangle whose base is circumference and whose height is vertical rise per turn.
Other useful calculators
Disclaimer: This spiral length calculator is for general educational and informational use only. It estimates curve length for an Archimedean spiral based on user-entered values and rounded output. It is not a substitute for professional engineering, manufacturing, fabrication, construction, safety, or installation review. Always verify final dimensions, tolerances, material behavior, bend radius, and project requirements before cutting or ordering material.
Last updated: May 21, 2026