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.
Example
For semi-axes a = 5 and b = 3:
| Property | Value |
|---|---|
| Area | π × 5 × 3 ≈ 47.124 |
| Perimeter | ≈ 25.527 |
| Focal distance c | √(25 − 9) = 4 |
| Eccentricity | 4 / 5 = 0.8 |
Everything updates instantly in your browser, and no input is uploaded.