Last updated: 2026-05-18

Ellipse

An ellipse is a closed, oval-shaped curve defined by two fixed points called foci (singular: focus). For any point on the ellipse, the sum of its distances to the two foci is constant. When both foci coincide at the center, the ellipse becomes a circle — so a circle is simply a special case of an ellipse.

Every ellipse has two axes of symmetry:

• Major axis: the longest diameter, passing through both foci. Half of it is the semi-major axis a.
• Minor axis: the shortest diameter, perpendicular to the major axis at the center. Half of it is the semi-minor axis b.

By convention, a ≥ b > 0. This calculator requires the semi-major axis to be at least as large as the semi-minor axis.

Area formula

The area of an ellipse is:

A = π × a × b

This exact result was first proved by Archimedes in his work Measurement of a Circle, using a method of exhaustion — essentially an early form of integration. The formula is beautifully simple: it generalizes the familiar circle area πr² (when a = b = r).

For example, an ellipse with a = 6 cm and b = 4 cm has:

A = π × 6 × 4 = 24π ≈ 75.40 cm²

Perimeter (circumference)

Unlike the area, there is no simple closed-form formula for the perimeter of an ellipse. The exact value is given by an elliptic integral — a type of integral that cannot be expressed in terms of elementary functions like square roots or trigonometry alone.

This calculator uses Ramanujan's first approximation (1914), which is remarkably accurate:

$P \approx \pi \times [3(a + b) - \sqrt{(3a + b)(a + 3b)}]$

For a circle (a = b = r):

$P = \pi \times [3(2r) - \sqrt{4r \times 4r}] = \pi \times [6r - 4r] = 2\pi r$ (exact)

For a typical ellipse the error is less than 0.0001%, making Ramanujan's formula accurate enough for any practical purpose.

Worked example

An ellipse with a = 6 cm and b = 4 cm:

$P \approx \pi \times [3(6 + 4) - \sqrt{(3 \times 6 + 4)(6 + 3 \times 4)}]$

$= \pi \times [30 - \sqrt{22 \times 18}]$

$= \pi \times [30 - \sqrt{396}]$

= π × [30 − 19.900...]
≈ π × 10.100
≈ 31.73 cm

Eccentricity

Eccentricity e measures how much the ellipse deviates from a perfect circle:

$e = \sqrt{1 - (b/a)^2}$

Eccentricity is always between 0 (inclusive) and 1 (exclusive) for an ellipse.

Real-world examples

Planetary orbits are ellipses with the Sun at one focus (Kepler's first law):

Earth's orbit is nearly circular (e ≈ 0.017), which is why our seasons are driven by axial tilt rather than distance from the Sun.

Athletic tracks are stadium ovals approximated by two semicircles joined by straight sections — the curved ends alone are semi-elliptical or semi-circular. A standard 400 m track has a total length of exactly 400 m.

Eggs look oval but are technically not ellipses: one end is more pointed than the other, making them asymmetric. A true ellipse is symmetric about both axes.