Ellipse Calculator
Calculate the area and perimeter of an ellipse from its semi-major and semi-minor axes.
How it works
- 1Area is computed exactly as π · a · b, where a is the semi-major axis and b is the semi-minor axis — a generalisation of the circle formula.
- 2Perimeter has no simple closed form, so the tool uses Ramanujan’s approximation π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))), where h = (a − b)² ÷ (a + b)².
- 3The diagram labels both semi-axes so you can see the shape the inputs describe.
Use cases
- Engineering and architecture: sizing oval windows, tunnels, or tank cross-sections.
- Astronomy: estimating orbital path lengths — planetary orbits are ellipses.
- Design and manufacturing: calculating material to cut or seal an elliptical border or gasket.
Frequently asked questions
Why is there no exact formula for the perimeter of an ellipse?
The perimeter is an elliptic integral that cannot be written with elementary functions. Ramanujan’s approximation is one of the most accurate simple formulas, with negligible error for most shapes.
What happens when a equals b?
When a = b the ellipse becomes a circle. The area (π·a·b → π·r²) and the perimeter approximation (→ 2π·r) reduce to the exact circle values.
Which axis is a and which is b?
By convention a is the semi-major (longer) axis and b is the semi-minor (shorter). The formulas are symmetric, so swapping them gives the same result.
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