What this calculator does
An ellipse is set by two radii at right angles: the semi-major axis, the longest distance from the centre to the edge, and the semi-minor axis, the shortest. A circle is the special case where the two are equal, which is why setting a and b equal here returns exactly the circle results.
Area is straightforward: π times the two axes multiplied together, a direct generalisation of πr². Perimeter is not; unlike a circle’s circumference, there is no exact elementary formula for it, so this calculator uses Ramanujan’s approximation, which is accurate to a fraction of a per cent for any ellipse.
The formula
Eccentricity measures how stretched the ellipse is, running from 0 for a perfect circle toward 1 for an infinitely flattened sliver. It comes from the ratio of the two axes: the more the minor axis falls short of the major axis, the closer eccentricity moves to 1. The focal distance is where the two focus points sit along the major axis, and it satisfies a² = b² + (focal distance)², the same Pythagorean relationship that defines the ellipse itself.
| Term | Meaning |
|---|---|
| a | Semi-major axis: the longest radius, centre to edge. |
| b | Semi-minor axis: the shortest radius, centre to edge. |
| Eccentricity | How elongated the ellipse is, from 0 (a circle) toward 1 (very flat). |
| Focus | One of two fixed points inside the ellipse whose distances to any point on the edge always sum to 2a. |
The inputs explained
| Field | What to enter |
|---|---|
| Semi-major axis a | The semi-major axis: half the longest diameter. |
| Semi-minor axis b | The semi-minor axis: half the shortest diameter. Enter the same value as a for a circle. |
When to use it
An oval garden bed, table or pool
Measured across its longest and shortest points, then halved to give the two semi-axes needed here, this calculator gives the area for materials and the perimeter for edging.
A satellite or planetary orbit
Orbits are ellipses with the larger body at one focus, not the centre. Eccentricity here is the same figure used in orbital mechanics to describe how far from circular an orbit is.
The elliptical shadow or cross-section of a cylinder
A pipe or tube cut at an angle produces an elliptical face. The semi-minor axis stays the pipe’s radius while the semi-major axis stretches with the cutting angle.
Camera aperture and lens design
An ellipse is the shape traced by a circle viewed at an angle, which is why eccentricity turns up in optics whenever a circular object is viewed obliquely rather than face-on.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
From a circle to a flattened ellipse, major axis fixed at 8
The longer axis held at 8 while the shorter axis shrinks from equal (a circle) down to a narrow sliver.
| Semi-minor axis | Area | Perimeter | Eccentricity |
|---|---|---|---|
| 1 | 25.133 | 32.726 | 0.9922 |
| 2 | 50.265 | 34.310 | 0.9682 |
| 4 | 100.531 | 38.754 | 0.8660 |
| 6 | 150.796 | 44.207 | 0.6614 |
| 8 | 201.062 | 50.265 | 0 |
Stretching an ellipse while the minor axis stays fixed
The shorter axis fixed at 4 while the longer axis extends outward.
| Semi-major axis | Area | Perimeter | Eccentricity |
|---|---|---|---|
| 4 | 50.265 | 25.133 | 0 |
| 6 | 75.398 | 31.731 | 0.7454 |
| 8 | 100.531 | 38.754 | 0.8660 |
| 12 | 150.796 | 53.458 | 0.9428 |
| 16 | 201.062 | 68.620 | 0.9682 |
Questions
Why is there no exact formula for the perimeter of an ellipse?
The perimeter involves an integral, technically an elliptic integral, named for exactly this problem, that has no closed form in elementary functions. Ramanujan’s formula, used here, approximates it to within about 0.04 per cent in the worst case.
What does an eccentricity of 0 mean?
A perfect circle. Eccentricity rises from 0 toward 1 as the ellipse becomes more elongated; a value close to 1 describes a very flat, cigar-shaped ellipse.
How is an ellipse different from an oval?
“Oval” is a general term for any egg- or ellipse-like closed curve. An ellipse specifically has two axes of symmetry and a precise mathematical definition; not every oval shape is a true ellipse.
How do I find the area if I only know the full width and height?
Halve both measurements to get the semi-major and semi-minor axes first: this calculator, like the standard formula, needs the radii, not the full diameters.
What are the foci used for?
They define the ellipse: any point on the edge has distances to the two foci that always add up to the same total, 2a. This property is what makes whispering galleries and satellite dish reflectors work.
For the special case of equal axes, see the circle calculator. For a wedge of a circle rather than an oval, use the sector and arc calculator.