Calculate the arc length of Archimedean, logarithmic, and Fermat spirals. Set parameters, number of turns, and view radius growth, pitch angle, and turn data.
Spirals are among the most beautiful and ubiquitous curves in nature and engineering. From the grooves on a vinyl record (Archimedean) to the shell of a nautilus (logarithmic) to the seed distribution in a sunflower head (Fermat), spirals appear at every scale and in every domain. Calculating the length of a spiral — its arc length — requires integration, since the curve has no straight segments.
This calculator computes the arc length of three major spiral types using numerical integration (Simpson's rule with high precision). The Archimedean spiral r = a + bθ grows at a constant rate, producing evenly spaced coils. The logarithmic (equiangular) spiral r = ae^(bθ) grows exponentially and is self-similar — every portion looks the same at every scale. Fermat's spiral r = a√θ grows as the square root of the angle, producing a pattern where successive coils get closer together.
Enter the parameters, select the spiral type, and set the number of turns to instantly see the total arc length, starting and ending radii, turn spacing, and — for logarithmic spirals — the constant pitch angle. Presets for real-world spirals let you explore immediately.
Computing spiral arc lengths by hand requires setting up and evaluating definite integrals — often involving hyperbolic functions or elliptic integrals that have no elementary closed form. This calculator uses high-precision numerical integration (Simpson's rule with adaptive step counts) to handle any parameter values and turn counts instantly.
It is perfect for engineers designing springs, scroll compressors, or antenna geometries; scientists studying phyllotaxis or galaxy morphology; and students exploring parametric curves and integration.
Arc length = ∫₀^θₘₐₓ √(r² + (dr/dθ)²) dθ. Archimedean: r = a + bθ, dr/dθ = b. Logarithmic: r = ae^(bθ), dr/dθ = abe^(bθ). Fermat: r = a√θ, dr/dθ = a/(2√θ). θₘₐₓ = 2πn where n = number of turns.
Result: Arc Length ≈ 1,005.6, End Radius = 31.42
An Archimedean spiral with a=0, b=0.5 over 10 turns (θ_max = 62.83 rad). The radius grows linearly from 0 to 31.42. The arc length is computed via numerical integration, yielding approximately 1,005.6 units.
The Archimedean spiral, with its constant radial spacing, is the basis for many engineering mechanisms. Record player grooves follow an approximately Archimedean path, as do the coils in flat spiral springs (like mainsprings in mechanical watches). In manufacturing, Archimedean spirals guide CNC toolpaths for spray coating and surface finishing, ensuring even coverage.
Jacob Bernoulli called the logarithmic spiral "Spira mirabilis" (wonderful spiral) because of its unique self-similarity: every portion of the curve is geometrically similar to every other. This property makes it ubiquitous in nature: nautilus shells, hurricane cloud bands, spiral galaxies, and even the flight patterns of certain insects all approximate logarithmic spirals. The constant pitch angle means that organisms growing along a logarithmic spiral maintain the same shape at every stage of development.
Fermat's spiral r = a√θ produces a tighter and tighter coil as θ increases (since successive turn spacings decrease as 1/√θ). When two Fermat spirals are plotted together — one for +√θ and one for −√θ — the resulting pattern of intersection points mimics the seed arrangement in sunflower heads. This phyllotactic pattern is mathematically optimal for packing seeds in a circular area, which is why evolution has converged on it repeatedly.
The arc length is the total distance along the spiral curve from the starting angle to the ending angle. It is computed via the integral ∫√(r² + (dr/dθ)²) dθ.
The Archimedean spiral (r = a + bθ) grows linearly — the spacing between successive turns is constant at 2πb. Use this as a practical reminder before finalizing the result.
The logarithmic spiral is self-similar: zooming in or out produces an identical curve. It also has a constant pitch angle — the angle between the tangent and the radial line is the same everywhere.
Fermat spirals describe the seed arrangement in sunflower heads and many other phyllotactic patterns in plants, optimizing packing density. Keep this note short and outcome-focused for reuse.
For Archimedean spirals, there is a closed form involving sinh⁻¹. For logarithmic spirals, L = (r_end − r_start)√(1 + 1/b²)/b. Fermat spirals require elliptic integrals. This calculator uses numerical integration for all types.
The pitch angle (for logarithmic spirals) is the constant angle between the tangent to the spiral and the radial direction. A smaller pitch angle means a tighter spiral; a larger angle means a more open one.