Ellipse Perimeter Calculator (Ramanujan)
P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))) where h = ((a−b)/(a+b))². Ramanujan approximation.
Result
Perimeter (≈ Ramanujan)
25.526999
Area = πab = 47.1239; eccentricity = 0.8000.
- a (semi-major)5
- b (semi-minor)3
- h = ((a−b)/(a+b))²0.062500
- Perimeter25.526999
- Area47.123890
- Eccentricity0.800000
Step-by-step
- h = ((5 − 3) / (5 + 3))² = 0.062500.
- P ≈ π(5 + 3) × (1 + 3h / (10 + √(4 − 3h))) = 25.5270.
- Area = π × a × b = 47.1239.
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
Perimeter requires the complete elliptic integral of the second kind. Numerical methods or approximations are the only options.
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