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An Archimedean spiral has the equation r = a + bθ, where the distance from the center grows steadily and evenly with the angle θ swept out — unlike a logarithmic spiral, which grows exponentially, an Archimedean spiral's coils are always evenly spaced apart, exactly like the grooves on a vinyl record or the coiled shape of a rolled-up garden hose or measuring tape. Its arc length between two angles is found via the standard arc-length integral for a polar curve, and this calculator applies it directly rather than relying on a memorized formula.
How it works
Enter the parameters a and b (from r = a + bθ) and the two bounding angles, in radians. Arc length along any polar curve r(θ) is given by the integral of √(r² + (dr/dθ)²) dθ; here, dr/dθ = b (a constant, since r is linear in θ), so the integrand becomes √((a+bθ)² + b²). The calculator numerically integrates this expression between your two angles using Simpson's rule with a large number of subintervals, since although an exact closed-form antiderivative does exist for this integral, it involves logarithms of the variable itself and is considerably more error-prone to implement correctly than a well-converged numerical approximation, which is accurate here well beyond ordinary display precision.
- Enter a (starting radius).
- Enter b (radius growth per radian).
- Enter θ1 (radians).
- Enter θ2 (radians).
- Click Calculate to see your results.
Examples
a=0, b=1, θ from 0 to 2π
This traces one full loop of the spiral r=θ. The arc length integral ∫√(θ²+1)dθ evaluated from 0 to 2π works out to approximately 21.26.
a=0, b=2, θ from 0 to π
This traces half a loop of the spiral r=2θ. The integral ∫√(4θ²+4)dθ = 2∫√(θ²+1)dθ from 0 to π evaluates to approximately 12.22.
a=5, b=0, θ from 0 to 2π
With b=0, the "spiral" degenerates into an ordinary circle of constant radius 5. The integral simplifies to ∫√(5²+0)dθ = 5×2π ≈ 31.42 — exactly matching the familiar circle circumference formula 2πr, a useful sanity check on the general formula.
Who should use it
- Estimating the length of material in a coiled hose, cable, or rolled sheet.
- Physics and engineering problems involving Archimedean spiral geometry.
- Calculus coursework on polar arc-length integrals.
Industry applications
- Mechanical engineering (springs, coils, cams)
- Calculus and physics education
Advantages
- Handles the general Archimedean spiral for any a, b, and angle range.
- Numerically robust, without relying on an error-prone closed-form antiderivative.
Limitations
- Only models the Archimedean (linear-growth) spiral family, not logarithmic or other spiral types.
Common mistakes to avoid
- Entering angles in degrees when the tool expects radians, which silently produces an incorrect arc length.
- Confusing an Archimedean spiral (evenly-spaced coils) with a logarithmic spiral (exponentially-widening coils) — they follow different equations and different arc-length formulas.
- Forgetting that a negative or zero b describes a degenerate or shrinking spiral rather than the typical expanding case.
Best practices
- Always double check that angles are entered in radians, not degrees, before submitting.
- Use the b=0 special case (which reduces to a simple circle) as a sanity check on your understanding of the formula.
- For very large angle ranges, expect a longer numerical integration but trust the result, since Simpson's rule remains accurate over broad domains for this smooth curve.
Tips
- Interested in a different classic rolling curve? See the Cycloid Calculator.