Ellipse
Find the area, perimeter, focal distance and eccentricity of an ellipse from its semi-axes.
The ellipse calculator finds the area, perimeter (circumference), focal distance, eccentricity and semi-latus rectum of an ellipse from its two semi-axes or from the semi-major axis and the focal distance.
What is an ellipse
An ellipse is a closed curve such that for each of its points the sum of the distances to two fixed points (the foci F₁ and F₂) is constant and equals 2a. Here a is the semi-major axis, b is the semi-minor axis, and c is the distance from the center to a focus.
Ellipse formulas
| Value | Formula |
|---|---|
| Area | S = πab |
| Perimeter (Ramanujan) | P ≈ π(3(a + b) − √((3a + b)(a + 3b))) |
| Focal distance | c = √(a² − b²) |
| Eccentricity | e = c / a, 0 ≤ e < 1 |
| Semi-latus rectum | p = b² / a |
| Equation | x²/a² + y²/b² = 1 |
Example. a = 5, b = 3.
S = π · 5 · 3 = 15π ≈ 47.12, c = √(25 − 9) = 4, e = 4/5 = 0.8, P ≈ 25.53.
The perimeter of an ellipse cannot be expressed in elementary functions — it is an elliptic integral. Ramanujan's formula has an error below 0.01% even for very elongated ellipses. If a = b, the ellipse is a circle.
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