Ellipse Eccentricity Calculator
From the two semi-axes, get the eccentricity, the focal distance, and the gap between the foci — the numbers that say exactly how round or how stretched an ellipse is.
Two axes, three answers
Enter the semi-major axis a and the semi-minor axis b and the calculator returns the eccentricity (c/a), the focal distance c, and the focus separation (2c) at once.
a must be the larger axis
The semi-major axis a has to be at least as long as the semi-minor axis b. If you enter a smaller value for a, the inputs are mislabelled and no result is shown.
What is an ellipse eccentricity calculator?
Two axes in, shape of the oval out
An ellipse eccentricity calculator turns the two half-widths of an ellipse — the semi-major axis a (the long one) and the semi-minor axis b (the short one) — into a single number that captures its shape: the eccentricity e. It runs from 0, a perfect circle, up toward 1, an ever longer and thinner oval. Alongside it the calculator finds the foci, the two special interior points that define the ellipse, by computing the focal distance c from the centre to each focus and the separation 2c between them. These appear everywhere from planetary orbits and satellite paths to optics, whispering galleries, and CAD.
Enter the semi-major and semi-minor axes to get the eccentricity, focal distance, and focus separation instantly.
Two short formulas, built from the two semi-axes a and b.
e = √(a² − b²) / aFirst find the focal distance c = √(a² − b²) — the distance from the centre to either focus, derived straight from the Pythagorean relation a² = b² + c². The eccentricity is then e = c / a, the focal distance as a fraction of the semi-major axis. The separation between the two foci is simply 2c.
Suppose an ellipse has a semi-major axis of 5 and a semi-minor axis of 3.
Focal distance
c = √(5² − 3²) = √(25 − 9) = √16 = 4 — centre to each focus.
Focus separation
2 × 4 = 8 — the distance between the two foci.
Eccentricity
e = c / a = 4 / 5 = 0.8 — a clearly elongated ellipse.
The eccentricity is the headline number, and it lives on a fixed scale from 0 to just under 1. At e = 0 the two axes are equal and the ellipse is a perfect circle, its foci collapsed onto the centre. As e grows the ellipse stretches: 0.8 (for a = 5, b = 3 here) is a distinctly oval shape, and values close to 1 are long, thin slivers. For comparison, Earth's orbit has an eccentricity of about 0.017 — so nearly circular that it looks round to the eye — while a comet can exceed 0.9. The focal distance c tells you how far each focus sits from the centre along the long axis, and the separation 2c is the gap between them; the larger they are relative to a, the more eccentric the shape. A handy property: every point on the ellipse has the same total distance (2a) to the two foci, which is why a loop of string pinned at the foci traces the curve. If your eccentricity comes out at exactly 0, double-check that you really meant a circle rather than mixing up which axis is the longer one.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Semi-axes, axis order, and consistent units
These formulas take the semi-axes (half-widths), not the full diameters — if you measured the full long and short widths, halve them first. The semi-major axis a must be the larger value; entering a smaller a than b is a mislabelled ellipse and returns no result, because a² − b² would be negative. The eccentricity itself is a pure ratio with no unit, but the focal distance and focus separation come back in whatever length unit you used for the axes, so keep one unit throughout.