Ellipse Perimeter Calculator

Work out ellipse perimeter instantly with clear inputs, formula shown and shareable results.

About Ellipse Perimeter Calculator

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.

Formulas

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))]

Worked example

With the default values below, Ellipse Perimeter Calculator returns Perimeter (Ramanujan II) of 41.3863. Change any input above and every figure here is recalculated in your browser.

Inputs used

Semi-major axis aminimum 0: 8
Semi-minor axis bminimum 0: 5

Results

Perimeter (Ramanujan II): 41.3863
Ramanujan I estimate: 41.3863
Naive circle estimate 2π(a+b)/2: 40.8407
Area: 125.6637

Step by step

1Values used
2Ramanujan II
3Ramanujan I
4Perimeter (Ramanujan II)= 41.3863
5Ramanujan I estimate= 41.3863
6Naive circle estimate 2π(a+b)/2= 40.8407
7Area= 125.6637

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How accurate is Ramanujan II?

For a/b up to about 3 the relative error is below 10⁻⁷, far better than the naive mean-radius circle estimate.

Why is there no exact formula?

The arc length integral is an elliptic integral of the second kind, which cannot be expressed with elementary functions.