How to Use the Ellipse Calculator
This calculator takes an ellipse's semi-major axis (a) and semi-minor axis (b) and instantly returns the three key values that describe its shape. Area is calculated exactly as π×a×b, an extension of the circle area formula (πr²), and is directly useful in architecture and landscaping for sizing elliptical plots or tracks.
Unlike a circle, an ellipse's perimeter cannot be expressed exactly with elementary functions — it's an elliptic integral. This calculator uses the Ramanujan approximation π[3(a+b)-√((3a+b)(a+3b))], which has an extremely small error and is accurate enough for practical use.
Eccentricity is calculated as √(1-b²/a²) and ranges from 0 (a perfect circle) to 1 (an extremely flattened ellipse), letting you see at a glance how circular or elongated the shape is. The semi-major axis must be greater than or equal to the semi-minor axis; if that condition isn't met or a value is 0 or below, the calculator immediately shows a guidance message.
Frequently Asked Questions
An ellipse's perimeter can only be expressed exactly using an elliptic integral, a transcendental function with no closed-form solution in elementary terms. This calculator uses the Ramanujan approximation, which has an extremely small error and is accurate enough for practical use.
Eccentricity (e) ranges from 0 (a perfect circle) to 1 (an extremely flattened, nearly straight-line ellipse). The closer e is to 0, the more circular the shape; the closer to 1, the more elongated it becomes.