Ellipse Calculator
Calculate the area, approximate circumference, eccentricity, and focal points of any ellipse. Enter the semi-major and semi-minor axes for instant, accurate results using Ramanujan's approximation.
An ellipse is a closed curve where the sum of distances from any point to two fixed points (foci) is constant. A circle is a special case of an ellipse where both axes are equal.
Ellipse Formulas:
- Area: A = pi x a x b, where a is the semi-major axis and b is the semi-minor axis.
- Circumference: There is no exact closed-form formula. Ramanujan's approximation is: C ≈ pi(a+b)(1 + 3h/(10 + sqrt(4 - 3h))), where h = ((a-b)/(a+b))^2.
- Eccentricity: e = c/a, where c = sqrt(a^2 - b^2). For a circle, e = 0. As the ellipse elongates, e approaches 1.
- Foci: Located at distance c from the center along the major axis.
Ellipses are fundamental in astronomy (planetary orbits are ellipses with the Sun at one focus, per Kepler's first law), optics (elliptical reflectors), architecture (whispering galleries), and engineering.