Ellipse Calculator
Calculate the area, circumference, eccentricity and focal points of an ellipse from its two semi-axes.
How to use this calculator
- 1Enter the semi-major axis — half the length of the longer diameter, measured from the centre.
- 2Enter the semi-minor axis, half the shorter diameter. If you enter them the other way round they are swapped automatically.
- 3Read the eccentricity to judge how elongated the shape is: 0 is a perfect circle.
How the calculation works
Area = π·a·b Circumference ≈ π(a+b)·(1 + 3h / (10 + √(4−3h))), where h = (a−b)²/(a+b)² e = √(1 − b²/a²) c = √(a² − b²)- a
- Semi-major axis — half the longer diameter
- b
- Semi-minor axis — half the shorter diameter
- e
- Eccentricity, from 0 (circle) toward 1 (increasingly flattened)
- c
- Distance from the centre to each focus
The area formula is exact and pleasingly simple: it is the circle formula πr² with the single radius replaced by the two semi-axes. Setting a = b recovers πr² exactly.
The circumference is not exact. An ellipse's perimeter is a complete elliptic integral of the second kind, which has no closed form in elementary functions. Ramanujan's 1914 second approximation, used here, has a relative error below roughly 3 × 10⁻⁵ across the whole range — accurate enough that the error is invisible at any practical precision.
Setting a = b makes eccentricity 0 and the focal distance 0, collapsing both foci onto the centre: a circle is an ellipse with zero eccentricity.
Worked example
An ellipse with semi-axes 5 cm and 3 cm
- 1.Area: π × 5 × 3 = 47.123890 cm².
- 2.For the circumference, h = (5 − 3)² ÷ (5 + 3)² = 4 ÷ 64 = 0.0625.
- 3.Circumference ≈ π × 8 × (1 + 3(0.0625) ÷ (10 + √(4 − 3(0.0625)))) = 25.526999 cm.
- 4.Eccentricity: √(1 − 9 ÷ 25) = √0.64 = 0.8.
- 5.Focal distance: √(25 − 9) = 4 cm, so the foci sit 4 cm either side of the centre.
Result: Area 47.1239 cm², circumference 25.5270 cm, eccentricity 0.8
What defines an ellipse
An ellipse is the set of all points where the distances to two fixed points — the foci — add up to the same constant. That constant is exactly the length of the major axis. This "two pins and a loop of string" definition is not just a construction trick; it is the actual mathematical definition, and it explains most of the shape's other properties.
When the two foci coincide, the constant-sum condition becomes "all points the same distance from one point", which is a circle. A circle is therefore not a different kind of shape from an ellipse — it is the special case where eccentricity is zero.
The circumference problem
The area of an ellipse is πab, an exact formula as clean as the circle's. The perimeter is a completely different story: it evaluates to a complete elliptic integral of the second kind, which provably cannot be written in terms of elementary functions. This is not a gap in anyone's cleverness — it is a proven impossibility, and it is where the entire field of elliptic integrals got its name.
So every practical ellipse circumference is an approximation. The crude π(a+b) is only accurate for near-circles. Ramanujan produced two approximations in 1914, the second of which — used here — holds a relative error below about three parts in a hundred thousand across the entire range of shapes, from circles to near-degenerate slivers. For any measurement or manufacturing purpose that is exact.
Eccentricity, and orbits
Eccentricity measures how far an ellipse departs from circular, running from 0 for a circle up toward 1 for something almost flat. Kepler's first law states that planets orbit the Sun in ellipses with the Sun at one focus — not at the centre, which is the detail most diagrams get visually wrong.
- Earth, e ≈ 0.017 — so nearly circular that a correctly drawn diagram looks like a circle. The Sun sits noticeably off-centre even so, which is what produces the small annual variation in distance.
- Mars, e ≈ 0.093 — visibly elliptical, and the reason its apparent brightness from Earth varies so much between oppositions.
- Halley's Comet, e ≈ 0.967 — extremely elongated — it swings inside Earth's orbit and then out past Neptune on the same path.
The reflective property
Any ray leaving one focus of an ellipse reflects off the boundary and passes exactly through the other focus. This single geometric fact has an unusual number of practical consequences: whispering galleries, where a murmur at one focus is audible at the other across a large hall; lithotripsy, which focuses shock waves generated outside the body onto a kidney stone placed at the second focus; and elliptical reflectors in optical and lighting systems that gather light from a source at one focus onto a target at the other.
What this assumes, and where it stops
Assumptions
- Both inputs are semi-axes (measured from the centre), not full axis lengths. Entering full diameters gives an ellipse twice the intended size.
- The axes are ordered so the semi-major is the larger; if entered the other way round they are swapped automatically.
Limitations
- The circumference is an approximation, not an exact value — no exact elementary formula exists. The error here is below about 0.003%.
- Covers ellipses only. Other conic sections (parabolas, hyperbolas) follow different formulas.
Common questions
Why is there no exact formula for an ellipse's circumference?
Because the arc length of an ellipse evaluates to a complete elliptic integral of the second kind, which has been proven not to be expressible in elementary functions. This is a genuine mathematical impossibility rather than an unsolved problem, which is why every practical formula — including the one used here — is an approximation.
What does eccentricity actually mean?
It measures how stretched the ellipse is, from 0 to just under 1. Zero means the two foci coincide and you have a circle. Values near 1 mean the foci are far apart relative to the size of the shape, producing a long thin ellipse. Earth's orbit has an eccentricity of about 0.017, which is why it looks circular.
Is a circle an ellipse?
Yes — a circle is the special case where both semi-axes are equal, eccentricity is 0 and both foci sit at the centre. Every ellipse formula reduces correctly: the area πab becomes πr², and the circumference approximation returns exactly 2πr.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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