Spiral Length Calculator

Calculate the arc length of an Archimedean spiral from inner radius, outer radius, and number of turns.

Enter Spiral Parameters

An Archimedean spiral has the equation r = a + bθ, where the spacing between turns is constant.

Result

Arc Length
0
units
Inner Radius (r1)1
Outer Radius (r2)10
Number of Turns5
Total Angle31.416 rad
Spiral constant a1
Spiral constant b0.28648
Spacing per turn1.8

Step-by-Step Solution

L = integral of sqrt(r^2 + (dr/dθ)^2) dθ

Archimedean Spiral Arc Length

An Archimedean spiral is a curve traced by a point moving away from a fixed center at a constant rate while rotating at a constant angular velocity. Its equation in polar coordinates is r = a + bθ, where a is the initial radius and b controls the spacing between turns.

Key Formulas

Spiral Equation

The polar equation of an Archimedean spiral.

r = a + bθ

Arc Length Integral

The exact arc length via integration.

L = ∫ sqrt(r² + b²) dθ

Spiral Constants

Derived from inner/outer radius and turns.

a = r1, b = (r2 - r1) / (2πN)

Approximation

For many turns, a good approximation is available.

L ≈ πN(r1 + r2)

How the Calculation Works

  1. From the inner radius, outer radius, and number of turns, determine the spiral constants a and b.
  2. The total angle swept is θmax = 2πN.
  3. The arc length integral is: L = ∫0θmax sqrt((a + bθ)² + b²) dθ.
  4. This integral has a closed-form solution using the formula for ∫ sqrt(u² + c²) du.

Applications

  • Clock springs: Calculating the length of flat coiled springs.
  • Vinyl records: Estimating the total groove length on a record.
  • Rolled materials: Paper rolls, wire coils, hose lengths.
  • Antenna design: Spiral antennas used in telecommunications.