Ellipse Circumference Calculator
Compute an ellipse's circumference using Ramanujan's high-accuracy approximation, along with its area and eccentricity, from the two semi-axes.
units
units
Ramanujan II approximation
- 1
h = (a − b)² ÷ (a + b)²
(5 − 3)² ÷ (5 + 3)² = 0.0625Ramanujan's shape parameter — 0 for a circle, higher for elongated ellipses. - 2
Correction term = 3h ÷ (10 + √(4 − 3h))
3 × 0.0625 ÷ (10 + √(4 − 3 × 0.0625)) = 0.015687 - 3
Circumference = π × (a + b) × (1 + term)
π × 8 × (1 + 0.015687) = 25.527
How does this calculator work?
Enter an ellipse's semi-major (a) and semi-minor (b) axes to get its circumference from Ramanujan's accurate approximation C ≈ π(a+b)(1+3h/(10+√(4−3h))), its exact area πab, and its eccentricity √(1−(b/a)²). Accurate to a fraction of a percent and exact for circles.
Formula
How this is calculated
Enter the semi-major axis a (half the longest diameter) and the semi-minor axis b (half the shortest diameter) in the same units. The calculator automatically treats the larger value as a and the smaller as b so the eccentricity is well defined, then reports the circumference, area, and eccentricity.
The perimeter of an ellipse has no simple closed form, so this uses Ramanujan's second approximation: with h = (a − b)² / (a + b)², the circumference is C ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))). This formula is accurate to within a tiny fraction of a percent for nearly all shapes, and is exact for a circle. The area is the exact value πab, and the eccentricity is e = √(1 − (b/a)²), ranging from 0 for a circle to nearly 1 for a very flattened ellipse.
Both axes must be positive. If a equals b the ellipse is a circle: h = 0, the circumference reduces to 2πa, and eccentricity is 0. Length outputs share the unit of the inputs; area is in those units squared and eccentricity is dimensionless.
Frequently asked questions
An ellipse perimeter is given by a complete elliptic integral with no elementary closed form. Ramanujan II is a simple expression that matches the true value to well under 0.01% across typical shapes.
No. The tool sorts your two inputs so the larger becomes the semi-major axis a and the smaller the semi-minor axis b before computing eccentricity, so the result is correct either way.
Eccentricity e measures how elongated the ellipse is: e = 0 is a perfect circle, and values approaching 1 describe increasingly stretched, flattened ovals.
Also known as
TG we-Calculate Editorial Team. (2026). Ellipse Circumference Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/ellipse-circumference-calculator
TG we-Calculate Editorial Team. "Ellipse Circumference Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/ellipse-circumference-calculator.
TG we-Calculate Editorial Team, "Ellipse Circumference Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/ellipse-circumference-calculator
@misc{wecalculate_ellipse_circumference_calculator, title = {Ellipse Circumference Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/ellipse-circumference-calculator}}, year = {2026}, note = {TG we-Calculate} }
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