Ellipse Calculator

About this calculator

An ellipse is defined by two radii — the semi-major axis (a) along its longest dimension and the semi-minor axis (b) along its shortest — and its area has an exact formula, π × a × b, a direct generalization of the circle's πr² (a circle is simply an ellipse where a = b). Its perimeter, however, has no closed-form elementary expression; instead it's computed with Ramanujan's remarkably accurate approximation, π[3(a+b) − √((3a+b)(a+3b))], which stays within a tiny fraction of a percent of the true value for almost any ratio of a to b.

Ellipses aren't just a textbook shape — planetary and satellite orbits are ellipses (Kepler's first law), architects use elliptical arches and domes for both structural and aesthetic reasons, and engineers design elliptical gears, mirrors and reflectors that exploit the shape's unique reflective geometry (a ray from one focus always reflects to the other). Landscapers and construction crews also use the area and perimeter formulas directly when laying out elliptical garden beds, pools or running tracks.

This calculator takes the two semi-axes of an ellipse and returns both its area and its perimeter using Ramanujan's approximation.