Ellipse Calculator

Area = pi x a x b. Perimeter uses Ramanujan's second approximation: pi x [3(a + b) - sqrt((3a + b)(a + 3b))], which stays within about 0.05% of the true value even for the most stretched ellipses. 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. All outputs are rounded to six decimal places.

The true perimeter of an ellipse requires a complete elliptic integral of the second kind, which cannot be expressed with elementary functions like square roots or logarithms. Ramanujan's second approximation keeps the relative error under 0.05% across the entire range of eccentricities, and it is far tighter for ellipses close to a circle, making it precise enough for engineering, architecture, and scientific work.

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.