Ellipse Perimeter Calculator
Unlike a circle, an ellipse has no exact elementary formula for its perimeter, which is a complete elliptic integral. The standard practical solution is Ramanujan's second approximation, accurate to a tiny fraction of a percent for typical shapes. Enter the semi-major and semi-minor axes (the half-widths in each direction) and this calculator returns the approximate perimeter, the exact area (pi times a times b), and the eccentricity, which measures how stretched the ellipse is. Use the same length unit for both axes.
Ellipse formulas
Let h = ((a - b) / (a + b)) squared
Perimeter = pi x (a + b) x (1 + 3h / (10 + square root of (4 - 3h)))
Area = pi x a x b
Eccentricity = square root of (1 - (b/a) squared), with a the larger axis
Focal distance 2c = 2 x square root of (a squared - b squared)
The perimeter uses Ramanujan's second approximation, the standard practical formula because no exact elementary expression exists. The area and eccentricity are exact.
Ellipse geometry context
- An ellipse perimeter is a complete elliptic integral with no elementary closed form.
- Ramanujan's second approximation is accurate to better than one part in ten thousand for typical shapes.
- The area is exactly pi times the two semi-axes.
- Eccentricity ranges from 0 (a circle) toward 1 (highly elongated).
- Enter half-widths (semi-axes), not full diameters.
Ellipse perimeter: frequently asked questions
How is the perimeter of an ellipse calculated?
An ellipse perimeter has no exact closed-form formula in elementary functions; it is a complete elliptic integral. This calculator uses Ramanujan's second approximation, which is accurate to a tiny fraction of a percent for typical ellipses. With h = ((a - b) / (a + b)) squared, P is approximately pi times (a + b) times (1 plus 3h divided by (10 plus the square root of (4 minus 3h))), where a and b are the semi-axes.
What are the semi-major and semi-minor axes?
The semi-major axis a is half of the longest diameter of the ellipse, and the semi-minor axis b is half of the shortest diameter, perpendicular to it. For a circle the two are equal. Enter both as half-widths, not full diameters.
How accurate is Ramanujan's approximation?
Very accurate. For a wide range of shapes the error is below one part in ten thousand, and it grows only for extremely elongated ellipses. It is the standard practical formula because the exact perimeter requires evaluating an elliptic integral that has no elementary closed form.
What is the area of an ellipse?
The area of an ellipse is exactly pi times the semi-major axis times the semi-minor axis: A = pi x a x b. Unlike the perimeter, the area has a simple exact formula. This calculator reports both the area and the eccentricity alongside the perimeter.
What is the eccentricity of an ellipse?
Eccentricity measures how stretched an ellipse is, from 0 for a circle up toward 1 for a very flattened ellipse. It equals the square root of (1 minus (b/a) squared) when a is the larger axis. A value near 0 is nearly circular; a value near 1 is highly elongated.
Official sources
- U.S. National Institute of Standards and Technology: DLMF Chapter 19, Elliptic Integrals.
- NASA: Orbital geometry and ellipses reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.