Conic Section Eccentricity Calculator

About this calculator

Eccentricity describes how much a conic section deviates from a perfect circle. For an ellipse with semi-major axis a and semi-minor axis b, e = √(1 − b²/a²), giving a value between 0 (a circle) and 1 (nearly a line); for a hyperbola, e = √(1 + b²/a²), always greater than 1. In both cases the focal distance from the center is c = a·e, which is what actually defines the position of the foci.

Astronomers and orbital mechanics engineers use eccentricity to characterize how elongated a planet's, comet's or satellite's orbit is — Earth's orbit is nearly circular at e ≈ 0.017 while many comets have e close to 1. Optical engineers rely on the same math when designing elliptical or hyperbolic reflectors and lenses, where a conic's eccentricity determines how it focuses light or radio waves.