Math
Ellipse Graph (x²/a² + y²/b² = 1)
Enter semi-axes a and b to graph the ellipse x²/a² + y²/b² = 1. Automatically computes area, perimeter, eccentricity, and foci. When a = b the ellipse becomes a circle.
Tips
- a is the semi-axis along the x-axis, b is along the y-axis. When a = b, the ellipse is a perfect circle.
- Eccentricity e measures how elongated the ellipse is. e = 0 means a perfect circle; values close to 1 produce a very elongated ellipse.
- Area is computed exactly as πab. The perimeter has no closed-form in elementary functions, so this tool uses the Ramanujan approximation: π(3(a+b) − √((3a+b)(a+3b))).
- Changing the display range does not change the ellipse shape. The foci (orange diamonds) are shown on the major axis.
FAQ
When a = b the ellipse becomes a perfect circle. Eccentricity is 0 and the two foci coincide at the centre.
The exact perimeter of an ellipse involves an elliptic integral that has no closed-form in elementary functions (except when a = b). This tool uses the Ramanujan approximation, which is accurate to within a fraction of a percent for typical shapes.
When a ≥ b the major axis is horizontal and the foci are at (±c, 0) where c = √(a²−b²). When b > a the major axis is vertical and the foci are at (0, ±c) where c = √(b²−a²). They appear as orange diamonds on the graph.
Side Note — Ellipses in planetary orbits
The word "ellipse" comes from the ancient Greek "ἔλλειψις" (elleipsis), meaning deficiency or falling short — when a cone is sliced at an angle, the cross-section falls short of a full circle.
Kepler's first law (1609) states that every planet orbits the Sun in an ellipse, with the Sun at one focus. Earth's orbital eccentricity is about 0.0167 (nearly circular), while Pluto's is about 0.249.