Ellipse Calculator: Foci, Vertices and Eccentricity

How it works

(x − h)²/rx² + (y − k)²/ry² = 1 · a = larger semi-axis, b = smaller · c² = a² − b² · foci at distance c from the center along the major axis · e = c/a · latus rectum 2b²/a · area πab

An ellipse is the set of points whose distances to two fixed points, the foci, add up to the same constant 2a (the length of the major axis). In standard form the denominators are the squares of the semi-axes; the larger one, a, sets the major axis and the direction in which the foci lie. The foci sit c units from the center with c² = a² − b², and the eccentricity e = c/a (between 0 for a circle and almost 1 for a very flat ellipse) measures how stretched it is. The perimeter has no elementary formula, so Ramanujan’s second approximation is used — accurate to many digits for ordinary ellipses.

Worked example

For (x − 2)²/25 + (y + 1)²/9 = 1 the center is (2, −1), a = 5 (horizontal) and b = 3. Then c = √(25 − 9) = 4, so the foci are (−2, −1); (6, −1), the vertices (−3, −1); (7, −1) and the co-vertices (2, −4); (2, 2). The eccentricity is 4/5 = 0.8, the latus rectum 2 × 9/5 = 3.6, the area 15π ≈ 47.1239 and the perimeter ≈ 25.527.

Sources: Ellipse — Wikipedia · OpenStax Precalculus 2e — The ellipse

Frequently asked questions