Ellipse Calculator
Calculate area, circumference, eccentricity and foci of an ellipse from its semi-axes.
About Ellipse Calculator
An ellipse is defined by its semi-major axis a and semi-minor axis b. The area is πab. The circumference has no elementary closed form; this uses Ramanujan's approximation. Eccentricity measures how elongated the ellipse is (0 = circle, approaching 1 = very flat).
Formulas
Area
A = π × a × b
- a
- Semi-major axis
- b
- Semi-minor axis
Eccentricity
e = √(1 − b²/a²)
- a
- Semi-major axis
- b
- Semi-minor axis
Worked example
With the default values below, Ellipse Calculator returns Area of 188.4956. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Semi-major axis (a)minimum 0 units
- 10 units
- Semi-minor axis (b)minimum 0 units
- 6 units
Results
- Area
- 188.4956 sq units
- Circumference (approx)
- 51.0540 units
- Eccentricity
- 0.800000
- Focal Distance (c)
- 8.0000 units
Step by step
- 1Area= 188.4956
- 2Eccentricity= 0.800000
- 3Focal distance from center= 8.0000
- 4Circumference (Ramanujan)= 51.0540
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What if a equals b?
Then the ellipse is a circle with radius a, eccentricity 0, and circumference 2πa.
How accurate is the circumference approximation?
Ramanujan's formula is accurate to within 0.05% for all eccentricities. The exact value requires an elliptic integral.