An ellipse is fully defined by just two numbers: its semi-major axis and its semi-minor axis. From those two values alone, you can derive the area, an accurate perimeter estimate, the eccentricity that describes its shape, and the location of its two foci. This calculator computes all four instantly and shows how they relate to each other — and to the familiar circle formulas you already know.

How the Ellipse Calculator works

Area follows directly from A = πab, a straightforward extension of the circle formula πr². Perimeter is trickier: no elementary formula exists for an ellipse's exact circumference, because it requires evaluating a complete elliptic integral of the second kind. Instead, this calculator uses Ramanujan's second approximation, P ≈ π(a+b)[1 + 3h/(10+√(4-3h))] where h = ((a-b)/(a+b))², which is accurate to within a small fraction of a percent for virtually any ellipse. Eccentricity, e = √(1 - b²/a²), and focal distance, c = √(a² - b²), both come directly from the Pythagorean relationship between a, b, and c in the ellipse's defining geometry.

Inputs and what they mean

Enter the semi-major axis (a) — half the ellipse's longest diameter — and the semi-minor axis (b) — half its shortest diameter. Both must be positive, and by definition a must be greater than or equal to b (if you have the axes reversed, just swap which value you enter where). The ratio between a and b, not their absolute size, is what determines eccentricity: a 10-by-6 ellipse and a 5-by-3 ellipse have the exact same eccentricity and shape, just scaled differently.

Limits and edge cases

When a = b, the ellipse becomes a circle — eccentricity is exactly 0 and the two foci merge at the center. The Ramanujan approximation used for perimeter is extremely accurate but not exact; for most engineering and design purposes, the tiny residual error (well under 0.1% for typical ellipses) is negligible. This calculator computes a standard axis-aligned ellipse centered at the origin — it does not model a rotated ellipse or ellipsoid (a 3D shape), which require different tools.