Ellipse Calculator

Area = pi x a x b. Perimeter uses Ramanujan's first approximation: pi x [3(a + b) - sqrt((3a + b)(a + 3b))]. Eccentricity = sqrt(1 - (b / a)^2), focal distance c = sqrt(a^2 - b^2), aspect ratio = a / b, and semi-latus rectum = b^2 / a.

The exact perimeter is expressed using a complete elliptic integral of the second kind, not elementary functions alone. This calculator uses Ramanujan's first approximation. It is exact for a circle and close for ordinary ellipses, but its relative error approaches about 0.42% as an ellipse becomes extremely elongated.

The semi-major axis a is half the longest diameter; the semi-minor axis b is half the shortest. By convention a >= b always holds. If you enter b larger than a, the calculator swaps the two values and flags the result as swapped, so the formulas stay correct.

Eccentricity measures how much the ellipse deviates from a circle. A value of 0 means a perfect circle; values approaching 1 describe a long, narrow ellipse. Every stretched ellipse falls strictly between 0 and 1. When a and b differ by less than 0.000000001, the calculator treats the shape as a circle and reports eccentricity as exactly 0.

Every ellipse has two foci on the major axis, each at distance c from the center, where c = sqrt(a^2 - b^2) = a x e. The defining property of an ellipse is that for any point on the curve, the sum of its distances to both foci equals exactly 2a. This relationship shows up in optics, orbital mechanics, and antenna design.