The area formula
A = πab
Where the formula comes from
An ellipse is a circle stretched in one direction. Stretching by a factor multiplies the area by the same factor, so a circle of radius b stretched to semi-axis a has area πb² × (a/b) = πab. When a equals b you get πr² back. The perimeter resists this treatment because stretching does not scale lengths uniformly, and the exact answer needs an elliptic integral.
A worked example, step by step
An ellipse with semi-axes of 5 cm and 3 cm
- Area: π × 5 × 3 = 47.12 cm².
- Ramanujan's approximation for the perimeter: 25.53 cm.
- Eccentricity: √(1 − 3²/5²) = 0.8.
- An eccentricity of 0 would be a circle.
Watch the units: area is always in square units.
Where it turns up in practice
Planetary orbits, which Kepler showed are ellipses with the Sun at a focus; whispering galleries and lithotripsy, both of which use the reflection property of the foci; and elliptical machine parts and arches. Eccentricity is how orbits are classified: Earth's is 0.017, almost circular.