Ellipse Perimeter Calculator — Ramanujan Approximation
The perimeter of an ellipse has no simple exact formula — this calculator uses Ramanujan's second approximation, accurate to within 0.02% for any ellipse, and also shows two simpler estimates for comparison.
Most accurate elementary approximation; error < 0.02% for any ellipse
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Shape parameter h
h = ((6 − 4) ÷ (6 + 4))² = 0.04 - 2
Correction denominator
10 + √(4 − 3 × 0.04) = 11.969772 - 3
Correction term
3 × 0.04 ÷ 11.969772 = 0.010025 - 4
Perimeter (Ramanujan 2nd)
π × (6 + 4) × (1 + 0.010025) = 31.7309
How does this calculator work?
The ellipse perimeter has no exact elementary formula. The best approximation is Ramanujan's: P ≈ π(a+b)[1 + 3h/(10+√(4−3h))] where h=((a−b)/(a+b))², accurate to 0.02% for any ellipse. For a circle (a=b), h=0 and P = 2πa exactly.
Formula
How this is calculated
Unlike the area of an ellipse (π × a × b, exact), the perimeter involves an elliptic integral of the second kind — a special function with no simple closed form. In practice, mathematicians use approximations. The three most common are: the simple estimate π√(2(a²+b²)), accurate to about 11% for very flat ellipses; Ramanujan's first formula π[3(a+b) − √((3a+b)(a+3b))], error under 1%; and Ramanujan's second formula π(a+b)[1 + 3h/(10+√(4−3h))], where h = ((a−b)/(a+b))², which is accurate to within 0.02% across the full range of eccentricities.
This calculator returns all three so you can see how they compare. For a circle (a = b) all three give the correct 2πr exactly (h = 0, so the correction term vanishes). As the ellipse becomes more elongated (eccentricity approaching 1), the simpler formulas drift while Ramanujan's second approximation stays very close to the true value.
Eccentricity e = √(a²−b²)/a ranges from 0 (circle) to 1 (degenerate flat ellipse). A value close to 1 means the ellipse is very elongated; near 0 it is nearly circular.
Frequently asked questions
The arc-length integral for an ellipse (x/a)²+(y/b)²=1 reduces to an "elliptic integral of the second kind", which cannot be expressed using only standard arithmetic, powers, and trig functions. It can be computed to any precision numerically, and Ramanujan's 1914 approximations are remarkably accurate elementary replacements.
For most practical purposes Ramanujan's second formula (the hero result above) is the best choice — it is accurate to better than 0.02% for any ellipse. If you only need a rough answer, the √(2(a²+b²)) formula is quick to compute by hand. Ramanujan's first formula is a good middle ground, accurate to within 1%.
Nothing — they are the same thing. Circumference is typically used for circles, while perimeter is more general and applies to any closed curve including ellipses. Both describe the total arc length of the boundary.
Also known as
TG we-Calculate Editorial Team. (2026). Ellipse Perimeter Calculator — Ramanujan Approximation [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/ellipse-perimeter-calculator
TG we-Calculate Editorial Team. "Ellipse Perimeter Calculator — Ramanujan Approximation." TG we-Calculate. 2026. https://we-calculate.com/calculator/ellipse-perimeter-calculator.
TG we-Calculate Editorial Team, "Ellipse Perimeter Calculator — Ramanujan Approximation," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/ellipse-perimeter-calculator
@misc{wecalculate_ellipse_perimeter_calculator, title = {Ellipse Perimeter Calculator — Ramanujan Approximation}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/ellipse-perimeter-calculator}}, year = {2026}, note = {TG we-Calculate} }
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