Ellipse Calculator

Area uses A = pi * a * b. Perimeter uses Ramanujan's second approximation, P is roughly pi * [3(a + b) - sqrt((3a + b)(a + 3b))], accurate to about 0.04 percent for any eccentricity. Eccentricity uses e = sqrt(1 - (b / a) squared), focal distance c = sqrt(a squared - b squared), aspect ratio = a / b, and semi-latus rectum p = b squared / a.

The exact perimeter of an ellipse is a complete elliptic integral of the second kind, which has no closed form in elementary functions. Ramanujan's second approximation gets you within roughly 0.04 percent across all valid shapes, which is more than enough for engineering, design, and most science work.

The semi-major axis a is half of the longest diameter, and the semi-minor axis b is half of the shortest. By convention a is always greater than or equal to b. If you enter b larger than a, the calculator swaps the two values and flags the result as swapped so you keep that convention.

Eccentricity measures how stretched the ellipse is. A value of 0 means the ellipse is a perfect circle, values close to 1 mean a long thin ellipse, and a value of exactly 1 would be a parabola. For any real ellipse, eccentricity sits between 0 and 1.

Every ellipse has two foci on its major axis. The focal distance c is the distance from the center to each focus, with c = sqrt(a squared - b squared) = a times e. The defining property of an ellipse is that for any point on the curve, the sum of its distances to the two foci equals 2a.