Calculate the area, perimeter, and eccentricity of an ellipse from its semi-major axis (a) and semi-minor axis (b).
An ellipse is a closed, oval-shaped curve where the sum of the distances from any point on the curve to two fixed points (called foci) is always constant. Every circle is a special case of an ellipse where both axes are equal. The longer axis is called the major axis and the shorter one the minor axis; this calculator uses their half-lengths, a (semi-major) and b (semi-minor).
Area = π × a × b
Perimeter (Ramanujan's 2nd approximation, accurate to within a fraction of a percent):
P ≈ π(a+b) × [1 + 3h / (10 + √(4 - 3h))], where h = [(a-b) / (a+b)]²
Eccentricity = √(1 - b²/a²) — a value between 0 (a circle) and 1 (a very flattened ellipse).
1. Enter the semi-major axis (a) — half the length of the longest diameter.
2. Enter the semi-minor axis (b) — half the length of the shortest diameter.
3. Choose your unit of measurement, then click Calculate.
For an ellipse with a = 6 cm and b = 4 cm:
Area = π × 6 × 4 = 75.40 cm²
h = [(6-4)/(6+4)]² = 0.04, so Perimeter ≈ 31.42 cm
Eccentricity = √(1 - 16/36) ≈ 0.745
Area = π × a × b (where a and b are the semi-major and semi-minor axes); circumference uses an approximation formula since there's no simple exact formula for an ellipse's perimeter.
A circle is a special case of an ellipse where both axes (a and b) are equal; an ellipse is "stretched" with different major and minor axis lengths.
Unlike a circle, an ellipse's perimeter has no simple closed-form formula and requires elliptic integrals for exact values, so calculators typically use accurate approximations like Ramanujan's formula.