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Spiral Length

Calculate the total length of an Archimedean spiral r = a + bθ. Input inner radius (a), spacing (b), and number of turns (n).

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Why: Understanding spiral length helps you make better, data-driven decisions.

How: Enter Inner radius (a), Spacing (b), Turns (n) to calculate results.

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ARCHIMEDEAN SPIRAL

Spiral Length — r = a + bθ

Inner radius (a), spacing (b), number of turns (n). Total arc length.

📐 Examples — Click to Load

Inputs

cm
cm

For educational and informational purposes only. Verify with a qualified professional.

📋 Key Takeaways

  • • Archimedean spiral: r = a + bθ — constant spacing between turns
  • • a = inner radius (at θ=0), b = spacing per radian, n = number of turns
  • • Arc length L = ∫√(r² + (dr/dθ)²) dθ from 0 to 2πn
  • • For a=0: L ≈ (b/2)[θ√(1+θ²) + ln(θ+√(1+θ²))] with θ = 2πn
  • • Applications: clock springs, vinyl grooves, spiral staircases, antennas

💡 Did You Know?

🌀Archimedes studied this spiral around 225 BC — one of the first curves beyond circle and lineSource: History
🪙Vinyl record grooves approximate Archimedean spirals — constant groove spacingSource: Audio
⏰Clock mainsprings use Archimedean spirals for even force deliverySource: Horology
📱Spiral antennas in phones use Archimedean shape for broadband performanceSource: RF Engineering
🏗️Spiral staircases often follow r = a + bθ for uniform step spacingSource: Architecture
🧬DNA double helix has a similar pitch — though not exactly ArchimedeanSource: Biology

📖 Formulas Explained

r = a + bθ

a = inner radius, b = spacing (increase in r per radian), θ = angle.

r=a+bthetar = a + b\\theta

Arc Length

L=int02pinsqrtr2+left(fracdrdthetaright)2,dtheta=int02pinsqrt(a+btheta)2+b2,dthetaL = \\int_0^{2\\pi n} \\sqrt{r^2 + \\left(\\frac{dr}{d\\theta}\\right)^2}\\,d\\theta = \\int_0^{2\\pi n} \\sqrt{(a+b\\theta)^2 + b^2}\\,d\\theta

🎯 Expert Tips

a = 0

When a=0, spiral starts at center. Common for art and math examples.

Large n

Vinyl records have n≈300+ turns. Use consistent units (e.g. b in cm/rad).

Units

a, b in same units. Length result in same units.

Spacing

Distance between turns = 2πb. So b = spacing/(2π).

⚖️ Comparison

Spiral TypeEquation
Archimedeanr = a + bθ (constant spacing)
Logarithmicr = ae^(bθ) (exponential)
Fermatr² = a²θ

📊 Stats

r
Radius
θ
Angle (rad)
2πb
Turn spacing
L
Length

❓ FAQ

What is an Archimedean spiral?

r = a + bθ. Constant spacing between turns. Named after Archimedes.

What is a (inner radius)?

Radius at θ=0. a=0 means spiral starts at origin.

What is b (spacing)?

Increase in r per radian. Distance between turns = 2πb.

Why radians?

Arc length integral uses radians. 1 turn = 2π radians.

Vinyl record?

Grooves approximate Archimedean. a≈6cm, b≈0.015cm, n≈300.

Clock spring?

Mainspring: a≈5mm, b≈1mm, n≈10. Length determines run time.

Logarithmic vs Archimedean?

Logarithmic: r=ae^(bθ), exponential growth. Archimedean: linear in θ.

Units?

Use same units for a, b. Result length in same units.

⚠️ Disclaimer: For ideal Archimedean spiral. Real-world spirals (vinyl, springs) may deviate. Educational use.

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