Ellipse Calculator

Area = π × a × b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). This generalises the circle area formula A = πr², which is the special case where a = b = r. For example, an ellipse with a = 6 cm and b = 4 cm has area = π × 6 × 4 = 75.40 cm².

There is no simple closed-form formula for the perimeter of an ellipse. The exact value involves an elliptic integral. The best practical approximation is Ramanujan's second formula: P ≈ π × [3(a+b) − √((3a+b)(a+3b))], which is accurate to within 0.0001% for all ellipses. Some simpler but less accurate approximations include P ≈ 2π√((a²+b²)/2).

Eccentricity (e) measures how stretched or elongated an ellipse is: e = √(1 − b²/a²), where a is the semi-major axis and b is the semi-minor axis. Eccentricity ranges from 0 (perfect circle, a = b) to just below 1 (extremely elongated, nearly a line). A value of 0.5 means a moderately oval shape. Earth's orbit has eccentricity ≈ 0.0167 (nearly circular).

An ellipse has two foci (singular: focus) located along the major axis at distance c = √(a² − b²) from the centre. For a horizontally oriented ellipse, the foci are at (±c, 0). A key defining property: for any point P on the ellipse, the sum of distances from P to both foci equals 2a. This property is used in whispering gallery architecture, satellite dish design, and orbital mechanics.

The major axis is the longest diameter of the ellipse, passing through both foci. Its length is 2a. The semi-major axis (a) is half of this - the distance from the centre to the farthest point on the ellipse. Similarly, the minor axis has length 2b, and the semi-minor axis b is its half. The semi-major axis is what you enter in the calculator.

A circle is a special case of an ellipse where the two axes are equal (a = b = r). In a circle, the two foci coincide at the centre, and eccentricity = 0. As a and b diverge, the ellipse becomes more elongated. Computationally, you can treat a circle as an ellipse with a = b - all ellipse formulas degenerate correctly to the circle formulas in that case.

The semi-latus rectum (l) is the half-length of the chord through one focus perpendicular to the major axis: l = b²/a. It is used extensively in orbital mechanics - for a planet orbiting the Sun, the semi-latus rectum determines the orbit's closest approach (perihelion) and farthest point (aphelion) in terms of the eccentricity.

The three key lengths of an ellipse satisfy the Pythagorean-like identity: a² = b² + c², or equivalently c² = a² − b². Here, c is the focal distance (distance from centre to each focus). This means a is always the hypotenuse of the right triangle formed by b, c, and a. You can rearrange to find any one given the other two: b = √(a² − c²), c = √(a² − b²).

Ellipses appear throughout science and engineering: planetary orbits (Kepler's first law), satellite and comet trajectories, whispering galleries (the Statuary Hall in the US Capitol), lithotripsy (shock wave therapy using elliptical reflectors), elliptical gears in bicycles, and optical mirrors in telescopes. The reflective property - that a ray from one focus reflects to the other - is the basis for many of these applications.

At e = 0.9, the ellipse is very elongated - more like a rugby ball or a narrow oval. Calculating: c = a × e = 0.9a, and b = a × √(1 − e²) = a × √(1 − 0.81) = 0.436a. So the minor axis is less than half the major axis in length. Comets that barely return from the outer solar system can have eccentricities of 0.99 or higher.