Ellipse Perimeter Calculator (Ramanujan)

P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))) where h = ((a−b)/(a+b))². Ramanujan approximation.

Inputs

Result

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How to use this calculator

  • Enter semi-major (a) and semi-minor (b).
  • Read perimeter using Ramanujan approximation.

About this calculator

Ellipse perimeter has no closed form (involves elliptic integrals). Ramanujan's second approximation P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))) is accurate to <1 ppm for typical eccentricities — extraordinary for a closed-form formula. For circles (a = b): h = 0, formula reduces to 2πr exactly. Ellipses appear in orbital mechanics (planet orbits), optics (focal points), and engineering (pressure vessels).

Frequently asked

Why no closed-form?+
Perimeter requires the complete elliptic integral of the second kind. Numerical methods or approximations are the only options.
Ramanujan accuracy?+
Less than 1 ppm error for h ≤ 0.7 (covers most ellipses). For extreme aspect ratios, alternative series exist.
a vs. b convention?+
a = semi-major axis (longer half-axis); b = semi-minor (shorter). The calc handles a < b without crashing.
Eccentricity?+
e = √(1 − b²/a²). 0 = circle. 1 = parabola (degenerate). Earth orbit: e ≈ 0.017.
Real-world use?+
Orbital tracks, oval running tracks, satellite paths, optical reflectors (whisper galleries), architectural arches.

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