The ellipse in analytic geometry

An ellipse is the locus of points in the plane whose sum of distances from two fixed points (the foci F₁, F₂) is constant and equal to 2a (the major-axis length). In canonical form centred at the origin with axes aligned to the coordinate axes: (x/a)² + (y/b)² = 1, where a is the semi-major axis and b the semi-minor axis (a ≥ b > 0). When a = b the ellipse degenerates to a circle of radius a. The ellipse is one of the three non-degenerate conic sections, obtained by intersecting a cone with a plane not parallel to its axis.

Area

The area enclosed by an ellipse is given by the exact formula A = π · a · b. This generalizes the area of a circle (A = π·r² is the case a = b = r). The cleanest proof uses a change of variables: x' = x/a, y' = y/b maps the ellipse onto the unit circle (area π); the inverse Jacobian is a·b, so the ellipse has area π·a·b. Example: a = 5, b = 3 gives A ≈ 47.12.

Perimeter (circumference)

Unlike the area, the perimeter has no elementary closed form. It is expressed through the complete elliptic integral of the second kind: P = 4a · E(e). In practice we use approximations. Simplest: P ≈ π·(a + b). Far more accurate is Ramanujan's second approximation: P ≈ π · (a + b) · [1 + 3h/(10 + √(4 − 3h))], with h = ((a − b)/(a + b))². Error below 0.0002% for a/b ≤ 10 and below 0.02% for a/b ≤ 100.

Eccentricity

Eccentricity measures how much the ellipse deviates from a circle: e = √(1 − b²/a²) = c/a, where c is the distance from centre to each focus. 0 ≤ e < 1: e = 0 is a circle, e → 1 an increasingly elongated ellipse. Earth's orbital eccentricity ≈ 0.0167 (nearly circular); Mercury 0.2056; Halley's comet ≈ 0.967 (highly elongated).

Foci and focal property

The foci F₁, F₂ lie on the major axis, symmetric about the centre, at distance c = a·e = √(a² − b²). The bifocal property: for every point P on the ellipse, |PF₁| + |PF₂| = 2a. A light ray emitted from one focus reflects off the ellipse to the other focus — used in whispering galleries and in shock-wave lithotripsy. The latus rectum (focal chord perpendicular to the major axis) has length 2b²/a.

Other parameters

• Semi-latus rectum ℓ = b²/a, related to angular momentum in Kepler orbits.
• Directrices: each directrix is a line at distance a/e from the centre; for every point on the ellipse, the ratio (distance to a focus / distance to the matching directrix) is constant and equals e.
• Sectorial area: with x = a·cos(θ), y = b·sin(θ), the area swept by the radius vector from angle 0 to θ is ½·a·b·θ.

Applications

Ellipses appear everywhere: planetary orbits (Kepler's first law — planets move on ellipses with the Sun at one focus); optics (ellipsoidal mirrors); acoustics (whispering galleries, e.g. the U.S. Capitol); architecture (St. Peter's Square is an ellipse ≈ 240 × 196 m); engineering (non-circular gears); naval design (sailing-boat hull sections).

How to use this calculator

Pick a mode: from semi-axes a, b (most common); from area and one semi-axis; from semi-major and eccentricity; from semi-axes and focal distance c. The tool returns: area (exact), perimeter (Ramanujan, ±0.0002%), eccentricity, focal distance c, latus rectum and semi-latus rectum. Use consistent units throughout. Positive IEEE-754 doubles accepted.

Limit cases

If a = b the ellipse is a circle: area π·a², perimeter 2π·a, eccentricity 0, foci coincide with centre. If b → 0 the ellipse degenerates into a segment of length 2a (zero area, perimeter 4a), eccentricity 1. For very elongated ellipses (e ≈ 1) Ramanujan remains accurate but numerical evaluation of E(e) gives maximum precision.

The ellipse perimeter and Ramanujan's formula

While the area of an ellipse has the elementary formula A = πab, the perimeter admits no closed form in elementary functions: its exact value is 4a·E(e), where E is the complete elliptic integral of the second kind — ellipses give these integrals their name. Approximations are therefore used. The most famous is Srinivasa Ramanujan's second formula (1914): P ≈ π(a+b)·(1 + 3h/(10 + √(4−3h))), with h = ((a−b)/(a+b))². Its accuracy is remarkable: the relative error stays below 10⁻⁵ even for highly elongated ellipses, and it is the formula implemented in this calculator. Ramanujan, a self-taught Indian mathematician, produced thousands of such results, many rigorously proved only decades later.

Foci, eccentricity and Kepler's orbits

An ellipse is the locus of points whose distances to two foci sum to a constant equal to 2a. The foci lie at c = √(a²−b²) from the centre, and the eccentricity e = c/a measures the flattening: it is 0 for a circle (a = b) and tends to 1 for an increasingly elongated ellipse. The latus rectum 2b²/a is the chord through a focus perpendicular to the major axis. Ellipses govern the cosmos: Kepler's first law (1609) states that planets orbit in ellipses with the Sun at one focus — Earth's orbit has eccentricity 0.0167, nearly circular, while Mercury's is 0.206. In acoustics, elliptical "whispering galleries" reflect sound from one focus to the other.

Updated July 2026