🏈 Ellipse Calculator
An ellipse is a stretched circle: the orbit of every planet, the shape of a whispering gallery, and the curve defined by two focal points — the sum of the distances to the foci is the same for every point on the curve. Its anatomy is the centre, the two semi-axes, the vertices and the foci.
Enter the centre and the two semi-axes, and this calculator returns the vertices, co-vertices, foci, eccentricity and area, with the ellipse and its foci drawn.
How to use this calculator
The key relation is c² = |a² − b²|, where c is the centre-to-focus distance along the major axis.
- Enter the centre (h, k) — leave 0, 0 for an ellipse centred on the origin.
- Enter a, the semi-axis along x, and b, the semi-axis along y (both positive).
- Press Calculate.
- Read the vertices, foci and eccentricity, and see the shape on the graph.
Ellipse Worked Example
An ellipse centred at the origin with a = 5 (along x) and b = 3 (along y): the major axis is horizontal with vertices (±5, 0), the co-vertices are (0, ±3), and c = √(25 − 9) = 4 puts the foci at (±4, 0). The eccentricity is e = 4/5 = 0.8 — a fairly stretched ellipse — and the area is πab = 15π ≈ 47.12. These are the defaults above.
Frequently asked questions
How do you find the foci of an ellipse?
Use c² = a² − b² with the larger semi-axis first: c is the distance from the centre to each focus, measured along the major axis. For a = 5, b = 3: c = √(25 − 9) = 4. The foci always sit inside the ellipse, on the major axis, symmetric about the centre.
What is eccentricity?
Eccentricity e = c / (major semi-axis) measures how stretched the ellipse is. e = 0 is a perfect circle; values close to 1 are long and thin. Planetary orbits are nearly circular (Earth ≈ 0.017) while Halley’s comet is about 0.97.
What is the difference between vertices and co-vertices?
Vertices are the endpoints of the major axis — the farthest points from the centre. Co-vertices are the endpoints of the minor axis — the nearest points on the curve. Together the four points frame the ellipse’s bounding box.
What is the equation of an ellipse?
(x − h)²/a² + (y − k)²/b² = 1, where (h, k) is the centre and a, b are the semi-axes along x and y. If a = b the equation becomes a circle of radius a — this calculator flags that special case.