Calculate area, circumference, eccentricity and foci of an ellipse from its semi-axes.
An ellipse is defined by its semi-major axis a and semi-minor axis b. The area is πab. The circumference has no elementary closed form; this uses Ramanujan's approximation. Eccentricity measures how elongated the ellipse is (0 = circle, approaching 1 = very flat).
Area
A = π × a × b
Eccentricity
e = √(1 − b²/a²)
Then the ellipse is a circle with radius a, eccentricity 0, and circumference 2πa.
Ramanujan's formula is accurate to within 0.05% for all eccentricities. The exact value requires an elliptic integral.