Find everything about an ellipse in one place. Enter the center and the two semi-axes, or paste the general equation, and this free ellipse calculator returns the standard form, foci, vertices, co-vertices, eccentricity, area, perimeter, latus rectum and directrices, then draws the graph.
For Ax² + Cy² + Dx + Ey + F = 0 (no xy term). Example: 9x² + 25y² − 225 = 0 → A = 9, C = 25, F = −225.
Ellipse formulas
Worked example
For x²/25 + y²/9 = 1, a = 5 and b = 3, so c = √(25 − 9) = 4. The foci are at (±4, 0), the vertices at (±5, 0), the eccentricity is 4/5 = 0.8 and the area is 15π ≈ 47.12. Its perimeter is about 25.53, slightly less than the circumference of a circle of radius 4.
There is no exact elementary formula for an ellipse’s perimeter, which is why this calculator uses Ramanujan’s second approximation. It is accurate to within a few parts per million for ordinary ellipses.
Frequently asked questions
How do I find the foci of an ellipse?
Work out c = √(a² − b²), where a is the semi-major axis and b the semi-minor axis. The foci lie on the major axis, c units either side of the center: (h ± c, k) for a horizontal ellipse or (h, k ± c) for a vertical one.
What is the eccentricity of an ellipse?
Eccentricity e = c / a measures how stretched an ellipse is. It is always between 0 and 1: e = 0 is a perfect circle and values close to 1 are long and thin. Earth’s orbit has e ≈ 0.017.
What is the formula for the area of an ellipse?
Area = π × a × b, using the two semi-axes. An ellipse with semi-axes 5 and 3 has area 15π ≈ 47.12 square units.
Is there an exact formula for the perimeter of an ellipse?
No simple exact formula exists; it needs an elliptic integral. Ramanujan’s approximation, used here, is extremely accurate for practical work.
Can I enter an ellipse in general form?
Yes. Choose General equation and enter A, C, D, E and F from Ax² + Cy² + Dx + Ey + F = 0. The calculator completes the square to find the center and semi-axes.
Related tools
Free educational tool by HawkInc. Results are calculated in your browser and rounded for display; check critical work by hand or with a second method.