Ellipse Calculator - Area, Perimeter, Eccentricity & Foci

Ellipse formulas

• Area: A = πab
• Perimeter (Ramanujan II): h = ((a − b)/(a + b))², P ≈ π(a + b)[1 + 3h/(10 + √(4 − 3h))]
• 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; the exact value is expressed with a complete elliptic integral. This calculator uses Ramanujan's second approximation: it first computes h = ((a − b)/(a + b))², then evaluates P ≈ π(a + b)[1 + 3h/(10 + √(4 − 3h))]. The approximation is exact for a circle and remains close for practical ellipses, but it is still an approximation. For very flattened ellipses or work needing tightly controlled precision, use a complete-elliptic-integral calculation.