Ellipse calculator
Area, approximate perimeter, eccentricity and foci of an ellipse from its semi-axes.
Fill in the fields and the result will appear here automatically.
An ellipse is specified by two positive semi-axes. Area S = πab, eccentricity and foci describe its geometry, while this tool estimates perimeter using Ramanujan’s first approximation. Its model error is separate from rounding and cannot be removed by showing more decimal places. Equal semi-axes give a circle; very elongated ellipses make the approximation less accurate. Semi-axis order does not matter: the larger is selected automatically.
How it works
Formula and logic
Area S = πab. The perimeter uses Ramanujan's approximation π[3(a+b) − √((3a+b)(a+3b))]. Eccentricity e = √(1 − b²/a²) is taken from the larger semi-axis, and the distance between the foci is 2√(a² − b²).
Example
An ellipse with semi-axes of 5 and 3 cm has an area of 47.124 cm², a perimeter of 25.527 cm and an eccentricity of 0.8.
Fields and units
- Length unit — list option
- Semi-axis a — cm
- Semi-axis b — cm
How to use
- — Choose the length unit.
- — Enter both semi-axes — halves of the axes, not the axes themselves.
- — Order does not matter: the larger one is detected.
- — Equal semi-axes give a circle.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- Ramanujan, 1914: first ellipse-perimeter approximation NIST DLMF: exact ellipse length using an elliptic integral OpenStax: area of a simple closed region from its boundary
- Limitation
- Area and focus formulas describe an ideal ellipse with a,b > 0; perimeter is approximate. For a nearly degenerate shape, rounded eccentricity may appear as 1. Selecting another unit relabels the numbers and does not convert them.
FAQ
Why is the perimeter approximate?
The exact perimeter is 4aE(e), with E the complete elliptic integral of the second kind. This tool keeps Ramanujan’s first formula. At a = 5, b = 3 it gives 25.5269864 instead of 25.5269989, about 0.00004885% relative error. At 99 and 1 the error is about 0.3414%. This formula has no universal 10⁻⁵% guarantee.
What does eccentricity tell me?
How stretched the ellipse is. Zero is a circle, values around 0.8 are visibly elongated, and approaching one means the shape is nearly a flat segment.
Is a semi-axis the same as an axis?
No, a semi-axis is half an axis: the distance from the centre to the edge, not edge to edge. Entering axes instead of semi-axes overstates the area fourfold.
What are the foci of an ellipse?
Two points on the major axis for which the sum of the distances to any point on the curve is constant. That property is what lets you draw an ellipse with a string looped around two pins.
What happens when the semi-axes are equal?
You get a circle: the area becomes πa², the eccentricity is zero and the foci meet at the centre. Ramanujan's approximation returns exactly 2πa there.