Ellipse Calculator
Enter the semi-major axis (a) and semi-minor axis (b) of an ellipse and instantly get the area, perimeter (using Ramanujan's highly accurate approximation), eccentricity, focal distance, latus rectum, and focal parameter. Switch between semi-axis and full-axis input, choose metric or imperial units, and see the step-by-step working.
Formula
Worked example
Ellipse with semi-major axis a = 5 m and semi-minor axis b = 3 m: Area = pi x 5 x 3 = 47.1239 m^2. h = (5-3)^2/(5+3)^2 = 4/64 = 0.0625. Perimeter = pi x 8 x [1 + 3(0.0625)/(10 + sqrt(4-0.1875))] = pi x 8 x 1.01836 = 25.6101 m. Eccentricity e = sqrt(1 - 9/25) = sqrt(0.64) = 0.8. Focal distance c = 5 x 0.8 = 4 m. Latus rectum = 2 x 9 / 5 = 3.6 m.
What is an ellipse?
An ellipse is a closed, symmetric curve in a plane that looks like a stretched or flattened circle. It is defined as the set of all points where the sum of distances from two fixed points (the foci) is constant. Every ellipse has a major axis (the longer diameter) and a minor axis (the shorter diameter), with their half-lengths called the semi-major axis (a) and semi-minor axis (b). When a equals b, the ellipse degenerates into a perfect circle. Ellipses appear throughout nature and engineering: planetary orbits follow elliptical paths (Kepler's first law), satellite dishes and whispering galleries use elliptical geometry, and many real-world oval shapes are well approximated by ellipses.
Ellipse area formula
The area of an ellipse is A = pi x a x b, where a is the semi-major axis and b is the semi-minor axis. This is a direct generalisation of the circle area formula (A = pi x r^2): when a = b = r, the result is pi x r x r = pi x r^2. The formula is exact and involves no approximation. For example, an ellipse with a = 5 m and b = 3 m has area pi x 5 x 3 = 47.1239 m^2.
Ellipse perimeter: why Ramanujan's formula?
Unlike area, the perimeter (circumference) of an ellipse has no simple closed form. The exact answer requires an infinite series of elliptic integrals. Ramanujan's 1914 approximation is the practical gold standard: P = pi(a+b)[1 + 3h/(10 + sqrt(4 - 3h))], where h = (a-b)^2/(a+b)^2. For a circle (a = b) h = 0 and the formula correctly gives P = 2pi*a. For all other ellipses the error is smaller than 0.0001%, far below measurement precision for any real-world use. An older simpler approximation P ≈ 2pi x sqrt((a^2 + b^2)/2) (sometimes called Euler's formula) is also used but is far less accurate, particularly at high eccentricities.
Eccentricity, foci, and other properties
Eccentricity (e) measures elongation: e = sqrt(1 - b^2/a^2), ranging from 0 (circle) to just below 1 (nearly flat). The focal distance c = ae gives how far each focus is from the centre; the two foci lie on the major axis symmetrically about the centre at positions (-c, 0) and (c, 0). The latus rectum L = 2b^2/a is the length of the chord that passes through a focus perpendicular to the major axis, and the focal parameter (semi-latus rectum) p = b^2/a is half that length. These quantities are important in orbital mechanics: for a planet orbiting the Sun, the Sun sits at one focus, the perihelion distance (closest approach) is a(1-e), and the aphelion distance (farthest point) is a(1+e).
Eccentricity and ellipse shape
| Eccentricity range | Shape description | Example |
|---|---|---|
| e = 0 | Perfect circle | a = b |
| 0 < e < 0.1 | Near-circular | Almost round |
| 0.1 - 0.5 | Mildly elongated | Oval shape |
| 0.5 - 0.85 | Moderately elongated | Notable oval |
| 0.85 - 0.99 | Highly elongated | Flat, needle-like |
| e approaches 1 | Degenerate (line) | b approaches 0 |
Eccentricity (e) ranges from 0 (perfect circle) to just below 1 (nearly flat line).
Frequently asked questions
What is the difference between the major and minor axes?
The major axis is the longest diameter of the ellipse, passing through both foci and the centre. Its half-length is the semi-major axis, labelled a. The minor axis is perpendicular to the major axis at the centre, and its half-length is the semi-minor axis, labelled b. By convention, a >= b. When a = b, the ellipse is a circle.
Why is there no exact formula for the perimeter of an ellipse?
The exact perimeter requires summing an infinite elliptic integral, which has no finite closed form in terms of standard functions. This was proven in the 19th century. Ramanujan's approximation (1914) is so accurate (error < 0.0001%) that it is the standard in all practical applications. This calculator uses that approximation.
What does eccentricity tell me about an ellipse?
Eccentricity (e) describes how "stretched" the ellipse is. A value of 0 means a perfect circle; values close to 1 mean a very flat, elongated ellipse. At exactly 1 the ellipse degenerates to a line segment. In astronomy, Earth's orbit has e = 0.0167 (nearly circular), while Halley's Comet has e = 0.967 (very elongated).
What are the foci of an ellipse?
The foci (singular: focus) are two special points inside the ellipse on the major axis, each at distance c = ae from the centre. For any point on the ellipse, the sum of its distances to both foci is constant and equals 2a (the full major axis length). This property makes ellipses useful for reflectors: a signal emitted from one focus is reflected through the other, which is the principle behind elliptical satellite dishes and acoustic whispering galleries.
What is the latus rectum?
The latus rectum is the chord of the ellipse that passes through a focus and is perpendicular to the major axis. Its length is L = 2b^2/a. It is useful in orbital mechanics: for a planet in an elliptical orbit around a star, the latus rectum determines the speed of the planet at the ends of this chord via the vis-viva equation.
Can I enter the full axis lengths instead of the semi-axes?
Yes. Switch the "Input mode" to "Full axes" and enter the full major axis (2a) and full minor axis (2b). The calculator divides each by 2 before computing, so the results are identical to entering the semi-axes directly.