Ellipse Calculator
Calculate ellipse area, perimeter, and eccentricity from semi-major and semi-minor axes.
Inputs
Results
Area
47.12
How to Use This Calculator
- Enter the Semi-Major Axis (a) — the longer half-axis — and the Semi-Minor Axis (b) — the shorter half-axis.
- Area = π × a × b; when a = b the ellipse becomes a circle with radius a.
- Perimeter uses the Ramanujan approximation: π × (3(a+b) − √((3a+b)(a+3b))), which is accurate to within 0.5%.
- Eccentricity e = √(1 − b²/a²) ranges from 0 (circle) to just under 1 (very elongated ellipse).
- The Ellipse Metrics chart plots the semi-major axis, semi-minor axis, area (÷π), and eccentricity side by side for quick comparison.
- Both axes must be positive; a must be ≥ b (swap them if needed).
How the result changes with Semi-Major Axis (a)
| Semi-Major Axis (a) | Area |
|---|---|
| 1,000 | 9,424.78 |
| 3,500 | 32,986.72 |
| 6,500 | 61,261.06 |
| 9,000 | 84,823 |
What each input means
- Semi-Major Axis (a)
- Length of the semi-major axis.
- Semi-Minor Axis (b)
- Length of the semi-minor axis.
How this is calculated
Formula
Area = πab, Perimeter ≈ π(3(a+b) - √((3a+b)(a+3b)))Worked example, using the default values
- Identify Input ParametersSemi-Major Axis (a) = 5, Semi-Minor Axis (b) = 3 = 2 input(s) provided
- Calculate AreaArea47.12 = 47.12
- Calculate PerimeterPerimeter25.53 = 25.53
- Calculate EccentricityEccentricity0.8 = 0.8
Engine last updated . Checked against 1 independently-derived test — how we verify calculators.
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