Ellipse Calculator

An ellipse is a closed curve where the sum of the distances from any point on the curve to two fixed interior points (the foci) is constant. It is the generalisation of a circle: a circle is simply an ellipse where both foci are at the same point. You define an ellipse using two measurements: the semi-major axis (a), which is half the longest diameter, and the semi-minor axis (b), which is half the shortest diameter. When a equals b you have a circle with radius a. When a is much larger than b the ellipse is long and narrow. This calculator returns four key properties. Area is pi times a times b, the exact formula. Circumference has no closed-form exact solution, so this calculator uses Ramanujan's highly accurate approximation: pi times (3(a plus b) minus the square root of ((3a plus b)(a plus 3b))). Eccentricity (e) measures how elongated the ellipse is, ranging from 0 for a circle to approaching 1 for a very flat shape; it is the square root of (1 minus b squared over a squared). The focal distance is how far each focus sits from the centre along the major axis, calculated as the square root of (a squared minus b squared). You enter a and b and receive all four values instantly. Ellipse calculations are used in astronomy (planetary orbits), optics (elliptical mirrors and lenses), engineering (stress analysis and cam design), and architecture. The semi-major axis must be greater than or equal to the semi-minor axis.

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Circumference uses Ramanujan's approximation, accurate to within 0.0001% for typical ellipses. The semi-major axis (a) must be greater than or equal to the semi-minor axis (b).