Case 1
- Semi-major axis a
- 5
- Semi-minor axis b
- 3
0.800000
Open with these valuesEllipse Eccentricity Calculator
0.800000
Result: 0.800000Eccentricity says how far an ellipse is from being a circle: e = √(a² − b²) ÷ a, a pure ratio between 0 and 1. Zero is a circle, values near one are long and thin. Both axes go in with the same length unit; the ratio itself has no unit.
The numbers at a glance
Held fixed: Semi-major axis a 5.000.
| Semi-minor axis b | Result |
|---|---|
| 1.000 | 0.979796 |
| 2.000 | 0.916515 |
| 3.000Your value | 0.800000 |
| 4.000 | 0.600000 |
| 5.000 | 0.000000 |
| 6.000 | 0.552771 |
Worked examples
Case 2
- Semi-major axis a
- 13
- Semi-minor axis b
- 5
0.923077
Open with these valuesCase 3
- Semi-major axis a
- 5
- Semi-minor axis b
- 5
0.000000
Open with these valuesHow it's calculated
e = √(a² − b²) ÷ a
- StepMeasure the long semi-axis a — half the longest diameter.
- StepMeasure the short semi-axis b, at a right angle to it.
- ResultRead the eccentricity as a number between 0 and 1.
Reference table
| a, b | Exact | Eccentricity |
|---|---|---|
| 5, 5 | 0 | 0.000000 |
| 5, 4 | 3/5 | 0.600000 |
| 5, 3 | 4/5 | 0.800000 |
| 10, 6 | 4/5 | 0.800000 |
| 2, 1 | √3/2 | 0.866025 |
| 13, 5 | 12/13 | 0.923077 |
Questions
How do you calculate the eccentricity of an ellipse?
Subtract the squared minor axis from the squared major axis, take the square root and divide by the major axis. With a = 5 and b = 3 that is 4/5, exactly 0.8.
What does an eccentricity of zero mean?
The two axes are equal, so the ellipse is a circle. The foci sit on top of each other in the centre.
Does it matter which axis I enter first?
No. The calculator takes the larger of the two as the major axis, so a swapped pair describes the same ellipse and gives the same answer.
Where are the foci?
At a distance of c = √(a² − b²) from the centre, on the long axis. With a = 5 and b = 3 that is 4, so the foci are 8 apart.
Which unit does the result use?
None. Eccentricity is a ratio of two lengths, so the units cancel out — as long as both axes use the same one.
Sources and last check
Information, not professional advice.