Frequently Asked Questions
Why isn't there a simple exact formula for an ellipse's perimeter?
What is Ramanujan's approximation for ellipse perimeter?
What is the difference between the semi-major and semi-minor axis?
How do I find the area of an ellipse?
Formula and Method for the Ellipse Perimeter
An ellipse is defined by two axes: the semi-major axis a (half the longest diameter) and the semi-minor axis b (half the shortest diameter). Unlike a circle's circumference (C = 2πr), an ellipse's perimeter has no elementary closed-form formula — the exact value is a complete elliptic integral of the second kind, E(e), where e is the eccentricity. This calculator instead uses Ramanujan's second approximation, published by Srinivasa Ramanujan in 1914, which gets within about 0.04% of the true value for any ratio of a to b and is exact when a = b (a circle).
How the calculation works
Enter the semi-major axis a and semi-minor axis b, and choose their unit. The calculator first computes h = ((a − b)/(a + b))², then applies P ≈ π(a + b)[1 + 3h/(10 + √(4 − 3h))]. It also reports the exact area A = πab, the eccentricity e = √(1 − (b/a)²) (0 for a circle, approaching 1 for a very flattened ellipse), and the distance between the two foci, 2c, where c = √(a² − b²) — the focal distance used in the "string and two pins" method of drawing an ellipse.
Common mistakes
- Using the full axes instead of the semi-axes: the formula needs a and b as half the major and minor diameters, not the full diameters — halve your measured width and height first.
- Assuming P = π(a + b) is exact: that simple average is only a rough estimate and can be off by several percent for elongated ellipses; Ramanujan's formula corrects for that with the h term.
- Confusing area and perimeter: the area (A = πab) is exact and easy, but the perimeter is fundamentally an approximation problem — don't expect the same precision from both.
Real-world applications
- Landscaping and construction use ellipse perimeter to estimate edging, fencing, or trim material for oval beds, tracks, or pools.
- Engineering and design use the eccentricity and focal distance to describe elliptical gears, orbits, arches, and optical reflectors.
- Astronomy uses the same semi-major/semi-minor axis relationship to describe planetary and satellite orbits, where the perimeter approximates the total orbital path length.