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Math & Statisticsgeometry

Ellipse Calculator

Ellipse area, perimeter and eccentricity. The area is exact and simple; the perimeter has no closed form at all, which is a genuinely surprising fact about a shape this ordinary.

Also called: oval calculator, ellipse area.

Area
47.12

Area 47.12, perimeter about 25.53. The eccentricity is 0.8, where zero is a circle and values near one are very elongated.

Perimeter (approximate)
25.53
Eccentricity
0.8
Distance from centre to each focus
4
On the perimeter
Approximate. An ellipse perimeter has no elementary closed form.
How this is calculated
Method and background

How this is calculated

Area is pi times the two semi-axes, a clean generalisation of the circle. The perimeter is not: it is an elliptic integral with no elementary closed form, so every published figure is an approximation. Ramanujan's second approximation is used here and is accurate to a few parts per million for any eccentricity you are likely to meet.

area is exact; the perimeter uses Ramanujan second approximation
a
Semi-major axis
b
Semi-minor axis

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a five by three ellipse

Semi-major axis
5
Semi-minor axis
3

Area47.12

pi * 15, and sqrt(1 - 9/25) = 0.8

Open this example

equal axes make a circle

Semi-major axis
5
Semi-minor axis
5

Area78.54

boundary: eccentricity zero and the exact circumference

Open this example

Method and limits

What it assumes

  • The semi-major axis is the larger of the two, though the arithmetic does not require it.

What it deliberately does not model

  • The perimeter is approximate by necessity rather than by choice.

Formula version 1.0.0 · definition 1.0.0 · India · Report a problem with this calculator

Frequently asked questions

Why is there no exact perimeter formula?
Because the arc length integral for an ellipse cannot be expressed in elementary functions. It defines a whole class of functions, the elliptic integrals, named after exactly this problem.