Ellipse Circumference Approximation Calculator
Estimate the perimeter of an ellipse when its semi-major and semi-minor axes are known. The result is an approximation, not an exact general ellipse formula.
Method, example, and limits
Formula: Ramanujan II: C ≈ π(a + b) [1 + 3h / (10 + √(4 − 3h))], where h = ((a − b)/(a + b))².
Inputs and units: a and b are the semi-major and semi-minor axis lengths in the same units; C is the estimated perimeter.
Worked example: For a = 5 m and b = 3 m, h = 0.0625 and the approximation gives C ≈ 25.526998 m (shown rounded to 25.527 m).
Assumptions: The axes are positive lengths for a planar ellipse. The formula is a high-accuracy approximation; it becomes exact for a circle, but still has a small approximation error for non-circular ellipses. This tool does not claim a universal error bound.
Primary source: S. Ramanujan, “Modular Equations and Approximations to π,” Quarterly Journal of Mathematics 45 (1914), 350–372.
Limits: If exact or certified bounds are required, use a numerical evaluation of the complete elliptic integral with an explicitly chosen error tolerance. The displayed result is rounded for readability.
Creator & reviewers
Creator persona
- Casey RowanUnits, conversions & measurement
Reviewer personas
- Devon ShawFormula & assumptions review
- Sage MitchellClarity & usability review