Ellipse calculator
Calculator
The calculator needs JavaScript. The tables below give the same numbers for common cases.
How it is worked out
- Area = π × a × b = π × 5 × 3 = 47.12389.
- There is no exact closed form for the perimeter; Ramanujan’s second approximation P ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))) with h = ((a − b) ÷ (a + b))² is accurate to many digits.
- Eccentricity e = √(1 − b² ÷ a²) = 0.8.
| Inputs | Answer |
|---|---|
| Semi-major axis a: 5; Semi-minor axis b: 3 | An ellipse with semi-axes 5 and 3 has area π × a × b = 47.12389 and perimeter about 25.526999. |
| Semi-major axis a: 10; Semi-minor axis b: 10 | An ellipse with semi-axes 10 and 10 has area π × a × b = 314.159265 and perimeter about 62.831853. |
| Semi-major axis a: 2; Semi-minor axis b: 1 | An ellipse with semi-axes 2 and 1 has area π × a × b = 6.283185 and perimeter about 9.688448. |
How to use it
The area of an ellipse with semi-axes a and b is π × a × b. The perimeter has no exact formula using ordinary functions; the calculator uses Ramanujan’s second approximation, which is accurate to many digits for any shape.
The eccentricity e = √(1 − b² ÷ a²) measures how stretched the ellipse is: 0 is a circle and values close to 1 are very elongated. The foci lie c = √(a² − b²) from the centre.
Frequently asked questions
What is the area of an ellipse?
π × a × b, where a and b are the semi-axes.
Why is the perimeter an approximation?
There is no simple exact formula; the exact value needs an elliptic integral.
What is the eccentricity of a circle?
0, since a = b.
Method: standard arithmetic shown step by step; no outside data is used. Checked 2026-11-12.
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