Work out ellipse perimeter instantly with clear inputs, formula shown and shareable results.
No elementary exact formula exists for an ellipse perimeter. Ramanujan's second approximation, P ≈ π(a+b)[1 + 3h/(10 + √(4 − 3h))] with h = ((a−b)/(a+b))², is accurate to better than one part in ten million for moderate eccentricity.
Ramanujan II
P ≈ π(a+b)[1 + 3h / (10 + √(4 − 3h))], h = ((a−b)/(a+b))²
Ramanujan I
P ≈ π[3(a+b) − √((3a+b)(a+3b))]
For a/b up to about 3 the relative error is below 10⁻⁷, far better than the naive mean-radius circle estimate.
The arc length integral is an elliptic integral of the second kind, which cannot be expressed with elementary functions.