Math · Geometry
Ellipse calculator.
Area, exact perimeter, eccentricity and foci from the two axes, with Ramanujan’s formula alongside.
Inputs
Ellipse
Enter your values.
How do you find the area and circumference of an ellipse?
An ellipse with semi-major axis a and semi-minor axis b has area πab. There is no simple closed form for the perimeter: it is 4a E(e), where E is the complete elliptic integral of the second kind and e = √(1 − b²/a²) is the eccentricity. This calculator evaluates that integral exactly with the arithmetic-geometric mean. Ramanujan’s approximations, π[3(a+b) − √((a+3b)(3a+b))] and a second, closer one, are shown for comparison. The foci sit c = √(a² − b²) from the centre.
Worked example. Semi-axes 10 and 6 give an area of 188.4956, a perimeter of 51.054 (Ramanujan II: 51.054), eccentricity 0.8 and foci 16 apart. A 2 by 1 ellipse has a perimeter of 9.688448; a circle (a = b) gives back 2πr.
Frequently asked questions
What is the circumference of an ellipse?
It has no exact elementary formula. The exact value is 4a E(e), an elliptic integral, and Ramanujan’s formula π[3(a+b) − √((a+3b)(3a+b))] is a very close approximation.
How accurate is Ramanujan’s formula?
Very. For semi-axes 10 and 6 the exact perimeter is 51.053998 and Ramanujan II gives 51.053998. The first formula is slightly less accurate for very elongated ellipses.
What is the area of an ellipse?
Area = π × a × b, where a and b are the semi-axes (half the full width and height). A circle is the special case a = b = r.
What is eccentricity?
e = √(1 − b²/a²). It is 0 for a circle and approaches 1 as the ellipse flattens.
Sources
Checked
Area, eccentricity, the perimeter p = 4aE(e) and the first Ramanujan formula are from the MathWorld Ellipse page; the second Ramanujan formula is eq. 71 there. The exact value is computed by the arithmetic-geometric mean method and checked in the tests against direct numerical integration. Runs entirely in your browser. Nothing you type is sent or stored.
Scope
Results are exact mathematics rounded for display, so they are only as good as the measurements you enter. Any length unit works as long as every value uses the same one; areas come out in that unit squared and volumes in that unit cubed. For homework, DIY layout and estimating, not a substitute for a surveyor, engineer or your teacher's required method.