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Triangular loop inductance calculator

When you need this

A loop shaped as an isosceles triangle: two sides and a base. The classic shape of a delta loop and of many home-made loop antennas.

Triangular loop: side a, base b, wire diameter dw
a — side, b — base of the loop, dw — wire diameter
Side a (m)
Base b (m)
Wire diameter dw (mm)
Permeability of the wire material μ
Result
Inductance L
The same in nanohenries

📐 Formulas​

A closed-form formula for a loop of three straight segments of round wire:

q = √(4·b²·a² − b⁴)
r = dw / 2

L [µH] = 0.2·μ · [ 2a·ln(2a/r) + b·ln(2a/r)
                     − 2(b + a)·arsinh(b²/q)
                     − 2a·arsinh((2a² − b²)/q)
                     − (2a + b) ]

Lengths are in metres; arsinh is the inverse hyperbolic sine.

The expression under the root is positive only for b less than 2a — which is the triangle inequality. The calculator checks it explicitly and gives a readable message instead of NaN.

Notation:​

  • a — side [m]
  • b — base of the loop [m]
  • dw — wire diameter [mm]
  • μ — relative permeability of the wire material
  • L — inductance [µH]
Limits of the calculator:
  • The base must be shorter than the sum of the sides: b less than 2a
  • A lumped-element model: the perimeter is much smaller than the wavelength
  • The wire is round and solid
  • The wire must not be too thick relative to the base

🎯 Where it is used​

  • The delta loop and other triangular loop antennas.
  • Coupling turns of an unusual shape.
  • Comparing shapes: for the same perimeter a triangle encloses less area than a square, so its inductance is lower too.
Practical notes:
  1. An equilateral loop with 1 m sides gives about 3.7 µH against 5.5 µH for a 1 × 1 m square — the difference is exactly the enclosed area.
  2. If the loop is very obtuse (the base almost twice the side), the formula still works, but the accuracy drops: check the result with a meter.
  3. For antennas it is not only the inductance that matters but the perimeter in wavelengths — work out both.

© 2026 UR3PKI · CyberDev.Space · Content licensed under CC BY-NC-SA 4.0.

How to cite this calculator
UR3PKI. «Triangular loop inductance calculator». CyberDev.Space. https://cyberdev.space/en/radio/calculators/coils/triangle_loop (licence CC BY-NC-SA 4.0).

The licence lets you use this material freely, including in teaching materials, but only with attribution to the author and a link to the source.

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