Archimedean Spiral Centerline Length Calculator
Calculate arc length of an outward Archimedean spiral using initial radius, radial pitch and number of turns.
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Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Parameterize the centerline with radius increasing linearly with angle. Integrate the polar arc-length integrand; report both cumulative arc length and radius by traveled turns.
r=R+(pitch/2π)θ; L=∫√(r²+(dr/dθ)²)dθ, θ from 0 to 2π×turns
Worked example
For r=θ over θ=0..1 radian, enter initial radius 0, pitch 2π length units and turns 1/(2π). Arc length is [√2+asinh(1)]/2 ≈1.147793575 length units.
Assumptions and input details
Outward spiral with nonnegative radius/pitch/turns. Pitch is centerline radius increase per full turn, not edge gap or diameter increase. Numerical adaptive quadrature; zero pitch counts repeated circular travel. No material width, self-contact, bend allowance or spring strength model. The graph shows radius versus turns, not an equal-scale physical drawing.
References: OpenStax: polar curve arc length
Unit definitions
Unit selection is independent of language. Each selector contains only units for its physical quantity. Calculations use defined base units internally.
Choose from 23 length definitions, including explicitly labelled regional measures. Area conversions square the length factor; geometric volumes cube it.
Regional measures are not interchangeable merely because their names look similar. This is a documented selection, not a claim to include every historical unit.
NIST measurement tables · Japanese traditional units · Mainland China and Hong Kong unit distinctions
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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