Math Calculators
Ellipse Calculator - Area, Perimeter, Eccentricity & Foci
Calculate ellipse area, perimeter (Ramanujan approximation), eccentricity, linear eccentricity, semi-latus rectum, and focal distance. Enter semi-major and semi-minor axes.
a must be ≥ b. For a circle, set a = b.
Quick examples:
Area47.12389
Perimeter (Ramanujan approx.)25.526986
Eccentricity (e)0.8
Linear eccentricity (c)4
Focal distance (2c)8
Semi-latus rectum (ℓ)1.8
Show calculation steps
- Semi-major axis a = 5
- Semi-minor axis b = 3
- Area = π·a·b = π·5·3 = 47.12389
- Perimeter ≈ π·[3(a+b) − √((3a+b)(a+3b))] = 25.526986
- Eccentricity e = √(1 − (b/a)²) = 0.8
- Linear eccentricity c = √(a²−b²) = 4
- Semi-latus rectum ℓ = b²/a = 1.8
- Distance between foci = 2c = 8
| Property | Value |
|---|---|
| Semi-major | 5 |
| Semi-minor | 3 |
Ellipse formulas
- Area: A = πab
- Perimeter (Ramanujan II): P ≈ π·[3(a+b) − √((3a+b)(a+3b))]
- Eccentricity: e = √(1 − b²/a²) (0 = circle, approaching 1 = flat)
- Linear eccentricity: c = √(a² − b²)
- Foci positions: (±c, 0) when major axis is horizontal
- Semi-latus rectum: ℓ = b²/a
Semi-axes explained
The semi-major axis (a) is half the longest diameter of the ellipse. The semi-minor axis (b) is half the shortest diameter. When a = b the ellipse degenerates to a circle.
Perimeter approximation
No exact closed-form formula for the ellipse perimeter exists using elementary functions. Ramanujan's second approximation is accurate to about 3 parts per million for all eccentricities, making it the standard choice for practical calculations.