This ellipse calculator finds an ellipse’s area, perimeter (circumference), focal distance, foci and eccentricity from its two semi-axes — with the honest note that ellipse perimeter has no simple exact formula.
Foci sit at (±c, 0) on the major axis, where c = √(a² − b²) — the two pin-points of the string-and-pins construction, and the reason planets orbit with the Sun at one focus (Kepler). Perimeter uses Ramanujan’s second approximation, accurate to ~1 part in 100,000 for typical ellipses: no closed-form elementary formula for it exists — the perimeter integral is literally why “elliptic integrals” have their name.
A Circle, Stretched — With Consequences
Set a = b and every formula collapses to the circle’s (that’s the Circle Calculator). Pull them apart and area stays graceful (A = πab) while perimeter becomes one of mathematics’ famous non-formulas, eccentricity e = c/a starts measuring the stretch (0 = circle, →1 = needle), and the foci acquire physical jobs from orbital mechanics to whispering galleries. Part of our Math Calculators collection.
Worked Example (Verified)
- a = 5, b = 3 (defaults): area = 15π ≈ 47.12 · perimeter ≈ 25.53 (Ramanujan) · c = √(25−9) = 4 · e = 0.8 — a noticeably stretched ellipse, foci at (±4, 0)
- Sanity limit: a = b = 5 → area 78.54, perimeter 31.42 — the circle formulas reappear exactly, as any honest generalization must reproduce
- Earth’s orbit: e ≈ 0.0167 — nearly circular, which surprises everyone expecting drama from the word “elliptical”
- Halley’s comet: e ≈ 0.967 — the needle end of the spectrum, a ≈ 17.8 AU against b ≈ 4.5 AU
Where Ellipses Earn Their Keep
- Orbits: Kepler’s first law — every orbit is an ellipse with the central mass at one focus.
- Acoustics/optics: rays from one focus reflect to the other — whispering galleries and lithotripters exploit it.
- Engineering: elliptical arches and tank heads distribute stress; cam design trades on the varying radius.
- The string construction: two pins at the foci + a loop of string = the gardener’s ellipse, length 2a + 2c of string around the pins.
Related Tools
- Circle Calculator — the e = 0 special case
- Area Calculator — areas across the shape zoo
- Math Calculators — the full collection
Frequently Asked Questions
How do you find the area of an ellipse?
A = π × a × b, with a and b the semi-major and semi-minor axes. A 5-by-3 ellipse (semi-axes) has area 15π ≈ 47.12 square units. If you’re given full axis lengths, halve them first — the most common ellipse error in the wild.
Why is there no exact perimeter formula?
The perimeter integral reduces to an elliptic integral of the second kind, which provably has no expression in elementary functions. Ramanujan’s approximation (used here) is accurate to roughly five significant figures for ordinary eccentricities — far beyond measurement precision for any physical ellipse.
What is eccentricity?
e = c ÷ a, where c = √(a² − b²) is the focal distance. e = 0 is a circle; values approaching 1 are needle-thin. Planetary orbits live near 0.01–0.2; comets push past 0.9. It’s the single number that says how “stretched” an ellipse is.
Where are the foci?
On the major axis, c units either side of center: (±c, 0) for a horizontal ellipse. Their defining property: for any point on the ellipse, the sum of distances to the two foci is constant (= 2a) — the fact the string construction, the reflection property and Kepler’s laws all stand on.
What’s the difference between major axis and semi-major axis?
Factor of two: the major axis is the full long diameter (2a); the semi-major is its half — the “radius” in the long direction. Formulas want the semi- axes. Orbital mechanics also quotes the semi-major axis as the orbit’s size and energy parameter.
Is this ellipse calculator free?
Yes — free, instant, no sign-up.