How to Use Spiral Length Calculator
The Spiral Length Calculator computes the arc length of an Archimedean spiral r = a + b·θ over a specified angular range. Enter the spiral parameters and angle limits, choose degrees or radians, and the calculator returns the arc length instantly. The Spiral Length Calculator runs entirely in your browser with no server calls.
- Enter the initial radius (a) — This is the radius of the spiral when θ = 0. It can be 0 for a spiral that starts at the origin.
- Enter the pitch parameter (b) — This controls how fast the radius grows per radian. Must be greater than 0.
- Select the angle unit — Choose degrees (°) or radians (rad).
- Set the start and end angles — Enter the starting and ending angles of the spiral arc you want to measure.
- Read the arc length — The Spiral Length Calculator shows the total length and the integral formula used.
The end angle must be greater than the start angle. If the pitch parameter b is zero or negative, the result is undefined.
Formula & Theory - Spiral Length Calculator
The Spiral Length Calculator uses the arc length integral formula for polar curves:
L = ∫[θ₀ to θ₁] √(r² + (dr/dθ)²) dθ
For Archimedean spiral r = a + b·θ, dr/dθ = b:
L = ∫[θ₀ to θ₁] √((a + b·θ)² + b²) dθ| Symbol | Meaning |
|---|---|
| r | Radial distance at angle θ |
| a | Initial radius (radius at θ = 0) |
| b | Pitch parameter (radius increase per radian) |
| θ₀ | Start angle (radians internally) |
| θ₁ | End angle (radians internally) |
The integral has no simple closed form in general, so the Spiral Length Calculator evaluates it numerically using Simpson’s rule with 1000 equally spaced steps, which provides high accuracy for smooth spirals.
Assumptions and Limits
The parameter b must be positive (the spiral must expand outward). Angles entered in degrees are converted to radians before integration. The result is in the same linear unit as the radius inputs (since only relative scale is computed).
Use Cases for Spiral Length Calculator
The Spiral Length Calculator is valuable whenever you need to know the actual path length along a spiral curve. Common uses include:
- Spring and coil design — Estimate the wire length needed to wind a flat spiral spring or coil.
- Scroll path planning — Calculate the travel distance of a milling head or pen along a spiral toolpath.
- Mathematics and education — Verify arc length homework problems involving polar curves and integrals.
- Rope and thread winding — Estimate how much material is used when winding material onto a spool in a flat spiral pattern.
The Spiral Length Calculator accepts any angle range, so you can compute just one turn, half a turn, or many turns of the spiral depending on your application.