Ellipse calculator
An ellipse is the oval traced by every point whose distances to two fixed foci add up to a constant. It is defined by two semi-axes: the semi-major axis a (half the longest width) and the semi-minor axis b (half the shortest). From just those two numbers, every other property follows. This calculator returns the area, perimeter, full axes, focal distance and eccentricity instantly.
How it works
Enter the two semi-axes in the same unit and the tool computes:
- Area — exact:
A = π · a · b. - Perimeter — there is no simple closed form, so it uses Ramanujan’s second approximation: with
h = (a − b)² / (a + b)²,P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))), accurate to many decimal places. - Focal distance —
c = √(a² − b²), the distance from the centre to each focus. - Eccentricity —
e = c / a, ranging from 0 (a circle) toward 1 (very elongated).
The major and minor axes are simply 2a and 2b. Order of input doesn’t matter — the larger value is taken as the semi-major axis.
Worked example
For semi-axes a = 5 and b = 3 (same units throughout):
| Property | Calculation | Result |
|---|---|---|
| Area | π × 5 × 3 | ≈ 47.124 |
| h for perimeter | (5−3)² / (5+3)² = 4/64 | 0.0625 |
| Perimeter (Ramanujan) | π(5+3)(1 + 3×0.0625 / (10+√(4−0.1875))) | ≈ 25.527 |
| Focal distance c | √(25 − 9) | 4 |
| Eccentricity e | 4 / 5 | 0.8 |
What eccentricity tells you
Eccentricity e measures how “stretched” the ellipse is on a scale from 0 to (but not including) 1:
- e = 0 — a perfect circle; the two foci coincide at the centre.
- e close to 0 — nearly circular, like Earth’s orbit around the Sun (roughly 0.017).
- e around 0.5 — visibly oval; the foci are noticeably off-centre.
- e close to 1 — very elongated, almost like a line; some comet orbits reach above 0.9.
In the example above, e = 0.8 means the ellipse is quite elongated — the focal distance (4) is 80% of the semi-major axis (5).
Why the perimeter has no exact formula
Unlike a circle, where C = 2πr, an ellipse perimeter involves an elliptic integral — an expression that cannot be simplified to elementary functions. Ramanujan’s second approximation (used here) is remarkable for compressing that integral into a formula accurate to better than one part in ten million for most practical shapes. Only at extreme eccentricities approaching 1 does the error grow measurably, and for any real-world engineering or design task the approximation is more than sufficient.
Common applications
- Garden and landscape design — laying out elliptical flower beds or paths by finding the foci and using the two-stakes-and-a-string method
- Architecture and acoustics — whispering galleries exploit the ellipse’s reflection property: a sound from one focus arrives at the other focus regardless of which path it takes
- Astronomy — Kepler’s first law states that planetary orbits are ellipses with the Sun at one focus; knowing a and b gives the semi-latus rectum and the range of distances from the Sun
- Engineering drawing — cutting a cylinder at an angle produces an elliptical cross-section; the cut angle and cylinder diameter determine the semi-axes