Compare multiple perimeter formulas, inspect approximation error, and visualize shape changes as your semi-major and semi-minor axes update.
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Axes input
cm
cm
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Results
Ellipse calculator results
Major and minor radii are semi-axes a and b (inputs may be swapped so a ≥ b). Circumference follows the formula you select in Formula explorer below. Curvature is the radius of curvature at the minor-axis vertex (ρ = a²/b). Foci: ±c on the major axis, c = √(a² − b²).
Note: Eccentricity and flattening describe how stretched the ellipse is versus a circle. Lower eccentricity means a rounder shape; values near 1 are very elongated.
This page calculates perimeter with multiple formulas, compares approximation error against an exact-series reference, and visualizes ellipse shape in real time.
Why ellipse perimeter needs approximations
Unlike a circle, an ellipse does not have a simple closed-form perimeter equation with only basic arithmetic. That is why calculators use high-quality approximations such as Ramanujan formulas, then compare them with a longer exact-series style reference.
What each output means
Circumference: perimeter from the selected formula tab.
Area: exact area from pi x a x b.
Eccentricity: how stretched the ellipse is (0 is a circle).
Flattening ratio: shape compactness from (a-b)/a.
How to choose a formula in practice
For most engineering, design, and classroom tasks, Ramanujan II is the best default because it balances speed and precision very well. The formula comparison table helps you quantify error versus the exact-series row before finalizing dimensions.
FAQ
Circle case?
If a = b, you get a circle circumference 2πa.
Exact formula?
Ellipse perimeter involves elliptic integrals—this uses a famous approximation.
Major/minor?
Semi-axes are half of the corresponding diameters.
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