Ellipse Calculator

Given the two half-axis lengths of an ellipse — the longer semi-major axis and the shorter semi-minor axis — this calculator returns the area (pi times one axis times the other), an approximate perimeter, and the eccentricity, a single number between 0 and 1 describing how stretched the shape is. Area treats both axes with perfectly equal weight — lengthening either one by ten percent grows the area by the identical proportional amount — but eccentricity treats them very differently and in opposite directions: stretching the semi-major axis while holding the semi-minor axis fixed makes the ellipse more elongated and pushes eccentricity up toward 1, while stretching the semi-minor axis while holding the semi-major axis fixed rounds the shape out and pushes eccentricity back down toward 0. When the two axes are equal, eccentricity comes out to exactly 0, because the ellipse has become a circle.

Perimeter has no simple closed-form formula for an ellipse in general, so the calculator uses a well-known approximation from the mathematician Ramanujan that stays close to the true value across ordinary axis ratios rather than evaluating an exact integral. One limitation worth naming: the calculator never checks that the semi-major axis you enter is actually the longer of the two — if the semi-minor axis field holds the larger value, the eccentricity formula ends up dividing a bigger number by a smaller one inside a square root of a negative quantity, and the result comes back as an undefined, non-numeric value instead of a warning.