Ellipse Calculator

An ellipse is a closed, oval-shaped curve defined as the set of all points where the sum of distances from two fixed points (called foci) is constant. It is a generalized case of a circle — when both foci coincide at the center, the ellipse becomes a circle. Ellipses appear in planetary orbits, engineering design, and many optical systems.

The area of an ellipse is calculated using the formula A = π × a × b, where 'a' is the semi-major axis and 'b' is the semi-minor axis. This is analogous to the circle area formula A = π × r², where both radii are equal.

Unlike a circle, there is no simple exact formula for an ellipse's perimeter. This calculator uses Ramanujan's second approximation: P ≈ π(a + b) × [1 + 3h / (10 + √(4 − 3h))], where h = ((a − b) / (a + b))². This is one of the most accurate approximations available.

Eccentricity (e) measures how much an ellipse deviates from a perfect circle. It is calculated as e = c / a, where c = √(a² − b²) is the linear eccentricity. Eccentricity ranges from 0 (a perfect circle) to just below 1 (a very elongated ellipse). The closer e is to 1, the more stretched the ellipse.

For a = 5 and b = 4, the linear eccentricity c = √(25 − 16) = √9 = 3. The eccentricity e = c / a = 3 / 5 = 0.6. This means the ellipse is moderately elongated, falling between a circle (e = 0) and a very flat ellipse (e close to 1).

The foci are two special points located along the major axis, each at a distance c = √(a² − b²) from the center. For an ellipse centered at the origin, the foci are at (−c, 0) and (+c, 0) if the major axis is horizontal. Any point on the ellipse has the property that the sum of its distances to both foci equals 2a.

The semi-latus rectum (ℓ) is the distance from a focus to the ellipse measured perpendicular to the major axis. It is calculated as ℓ = b² / a. It appears in orbital mechanics and describes the shape of a Keplerian orbit.

To find the area of an oval, identify its two perpendicular radii — the semi-major axis (a) and semi-minor axis (b) — then apply the formula A = π × a × b. For example, an oval with a = 10 and b = 6 has an area of π × 10 × 6 ≈ 188.50 square units.