About the ellipse calculator
An ellipse is a stretched circle with semi-axes a (the long half-width) and b (the short one). Its area is simply πab, but its perimeter has no elementary formula. Most calculators use Ramanujan’s approximation; this one computes the exact perimeter to full double precision using the arithmetic–geometric mean.
The foci sit a distance c = √(a² − b²) from the center, and the eccentricity e = c/a measures how far the ellipse is from being a circle (e = 0).
Formulas
| Area | K = πab |
| Perimeter | P = 4a·E(e), computed exactly (AGM method). Ramanujan’s approximation: P ≈ π[3(a+b) − √((3a+b)(a+3b))] |
| Focal distance | c = √(a² − b²) |
| Eccentricity | e = c/a |
Frequently asked questions
Is there an exact formula for the perimeter of an ellipse?
Not in elementary functions, it is a complete elliptic integral of the second kind. It can still be computed to any precision; this calculator uses the fast-converging AGM series.
What is eccentricity?
e = c/a = √(1 − b²/a²). A circle has e = 0; very flat ellipses approach e = 1.