Ellipse Area, Circumference & Eccentricity Calculator

An ellipse's area has a simple exact formula: area = π × a × b, where a and b are the lengths of its semi-major (longer) and semi-minor (shorter) axes — the direct generalization of a circle's πr², since a circle is just an ellipse with a = b.

Its circumference has no elementary exact formula at all — it's an elliptic integral. This calculator uses Ramanujan's second approximation, one of the most accurate named approximations in common use: C ≈ π(a+b) × (1 + 3h / (10 + √(4 − 3h))), where h = ((a−b)/(a+b))². It's exact when a = b (reducing to the circle's 2πr) and stays accurate to within a few parts in a million even for visibly elongated ellipses.

Eccentricity, e = √(1 − (b/a)²), measures how far the ellipse departs from a circle: e = 0 is a perfect circle, and e approaching 1 describes an increasingly elongated, cigar-shaped ellipse. Axis lengths are treated as plain numbers, not a physical measurement with a unit to convert — the result comes out in whatever unit the two axes were entered in, the same as this site's circle and triangle calculators.