Case 1
- Semi-major axis a
- 5
- Semi-minor axis b
- 3
25.526986
Open with these valuesEllipse Perimeter Calculator
25.526986units
Result: 25.526986 unitsAn ellipse has no exact perimeter formula in elementary functions, so this uses Ramanujan's first approximation — accurate far beyond the six digits shown. Both semi-axes go in with the same length unit and the perimeter comes back in it. Equal axes give the circle, 2πr.
The numbers at a glance
Held fixed: Semi-major axis a 5.000.
| Semi-minor axis b | Result |
|---|---|
| 1.000 | 21.005604 |
| 2.000 | 23.012812 |
| 3.000Your value | 25.526986 |
| 4.000 | 28.361668 |
| 5.000 | 31.415927 |
| 6.000 | 34.628956 |
Worked examples
Case 2
- Semi-major axis a
- 10
- Semi-minor axis b
- 5
48.442105
Open with these valuesCase 3
- Semi-major axis a
- 5
- Semi-minor axis b
- 5
31.415927
Open with these valuesHow it's calculated
P ≈ π(3(a + b) − √((3a + b)(a + 3b)))
- StepMeasure the long semi-axis a — half the longest diameter.
- StepMeasure the short semi-axis b, at a right angle to it.
- ResultRead the perimeter in the same unit you entered.
Reference table
| a, b | Note | Perimeter |
|---|---|---|
| 2.5, 2.5 | circle, 5π | 15.707963 |
| 4, 2 | 2:1 ellipse | 19.376842 |
| 5, 3 | Ramanujan | 25.526986 |
| 5, 5 | circle, 10π | 31.415927 |
| 10, 5 | 2:1 ellipse | 48.442105 |
Questions
How do you calculate the perimeter of an ellipse?
There is no exact elementary formula, so a very close approximation is used: π(3(a + b) − √((3a + b)(a + 3b))). With a = 5 and b = 3 it gives about 25.526986.
How accurate is Ramanujan's approximation?
For ordinary ellipses the relative error stays below one part in a billion. It is far tighter than the six decimals shown here.
Why is there no exact formula?
The perimeter is an elliptic integral, which cannot be written with roots, logarithms and trigonometric functions alone. That is what gave elliptic integrals their name.
What happens when both axes are equal?
The ellipse is a circle and the formula reduces exactly to 2πr. Axes of 5 give 10π, about 31.415927.
Sources and last check
Information, not professional advice.