Helical Coil Calculator — Wire Length & Inductance
Find the total wire length (exact helix arc length) and inductance of a single-layer air-core helical coil — enter number of turns, mean coil diameter and coil length.
mm
mm
L = N × √((π·D)² + pitch²) — exact helix arc length
- 1
Pitch (axial spacing per turn)
pitch = 50 ÷ 50 = 1 mm - 2
Circumference per turn (πD)
C = π × 20 = 62.83 mm - 3
Wire length per turn
√(62.83² + 1²) = 62.84 mmPythagorean combination of the circumference and axial pitch — exact helix arc length per turn. - 4
Total wire length
50 × 62.84 ÷ 1000 = 3.142
How does this calculator work?
Total wire length = N × √((πD)² + (ℓ/N)²) — the exact helix arc length for N turns, mean diameter D and coil length ℓ. Inductance ≈ μ₀N²π(D/2)²/ℓ (solenoid formula; most accurate when ℓ/D > 1). All lengths in consistent units; wire diameter does not affect these results.
Formula
How this is calculated
A helical coil (solenoid) is formed by winding N turns of wire around a cylinder of mean diameter D over an axial length ℓ. The spacing between adjacent turns measured along the axis is the pitch p = ℓ/N.
The exact wire length follows from the arc-length formula for a helix: each single turn traces a path whose length is √((π·D)² + p²) — the Pythagorean combination of the circumference per turn (π·D) and the axial advance per turn (the pitch p). Multiplying by N gives the total wire length. This formula is exact for an ideal helix and does not depend on the wire diameter or winding tension.
For inductance, the calculator uses the ideal solenoid formula L = μ₀·N²·A/ℓ, where A = π·(D/2)² is the cross-sectional area of the coil and μ₀ = 4π × 10⁻⁷ H/m is the permeability of free space. This formula is accurate for long coils (aspect ratio ℓ/D > 0.5–1). For short, fat coils it underestimates inductance; Wheeler's 1928 empirical formula or the Nagaoka coefficient correction should be used instead. A warning appears when the aspect ratio falls below 0.5.
Frequently asked questions
N × πD gives only the horizontal circumference contribution, ignoring the axial advance. Because the wire travels both around the cylinder and along its axis, the path per turn is a hypotenuse: √((πD)² + pitch²). For tightly wound coils the pitch is small and the difference is tiny, but for loosely wound coils it becomes significant.
The solenoid formula L = μ₀N²A/ℓ is exact for an infinitely long solenoid. For practical coils, it is a good approximation when ℓ/D > 1. For short, wide coils the formula underestimates inductance by 10–30%. Wheeler's 1928 formula L [μH] ≈ 0.394·N²·r² / (9r + 10ℓ) (r, ℓ in cm) is more accurate for the full range.
The helix angle α is the angle between the wire and the plane perpendicular to the coil axis: α = arctan(pitch/(π·D)). A helix angle of 0° means horizontal turns; a large angle means the wire spirals steeply along the axis. Most practical coils have helix angles below 10°.
Also known as
TG we-Calculate Editorial Team. (2026). Helical Coil Calculator — Wire Length & Inductance [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/helical-coil-calculator
TG we-Calculate Editorial Team. "Helical Coil Calculator — Wire Length & Inductance." TG we-Calculate. 2026. https://we-calculate.com/calculator/helical-coil-calculator.
TG we-Calculate Editorial Team, "Helical Coil Calculator — Wire Length & Inductance," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/helical-coil-calculator
@misc{wecalculate_helical_coil_calculator, title = {Helical Coil Calculator — Wire Length & Inductance}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/helical-coil-calculator}}, year = {2026}, note = {TG we-Calculate} }
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