Ellipse Calculator

Q: How do you find the area of an ellipse?

Multiply pi by the two semi-axes: A equals pi a b. It is the generalisation of a circle's pi r squared, since a circle is an ellipse with equal axes. Semi-axes 5 and 3 give an area of about 47.12.

Q: How do you find the circumference of an ellipse?

There is no exact elementary formula; the true circumference is an elliptic integral. In practice a close approximation is used. This calculator uses Ramanujan's formula, accurate to many decimal places for any realistic ellipse.

Q: What is the difference between the major and minor axis?

The major axis is the longest diameter; the minor axis is the shortest, at right angles to it. Semi means half, so the semi-major and semi-minor axes run from the centre to the edge along each direction.

Q: Is a circle an ellipse?

Yes. A circle is the special case where the two semi-axes are equal. Its eccentricity is zero, both foci sit at the centre, and the circumference reduces to 2 pi r.

An ellipse is a stretched circle, set by two radii: the semi-major axis (the longer one) and the semi-minor axis (the shorter). Enter both and get the area, circumference, eccentricity and foci, drawn to scale. You can also click a value on the drawing to change it.

Semi-axes

a & b

Enter both semi-axes — the horizontal (a) and vertical (b) radii.

Semi-axis a (horizontal)

Semi-axis b (vertical)

Units Decimals

Click an amber value to edit it · circumference is approximate

Results

Ellipse formulas

With semi-axes a and b, the area is exact and simple. The circumference has no exact elementary formula, so this tool uses Ramanujan's approximation, which is accurate to far more decimals than you'll ever need.

Area · A = π a b

Eccentricity · e = √(1 − b²/a²)

Focal distance · c = √(a² − b²)

The foci

Every ellipse has two foci on its major axis, each a distance c from the centre. For any point on the curve, the distances to the two foci always add to the same total — that constant sum is what defines an ellipse.

When a = b

If the two semi-axes are equal, the ellipse is a circle: the eccentricity is zero, the foci meet at the centre, and the circumference becomes the familiar 2πr.

Working with ellipses

An ellipse is what you get when you stretch a circle along one direction — the shape of a planet's orbit, a tilted circle seen in perspective, or a running track's curve. It's defined by two radii measured from the centre: the semi-major axis (the longer one) and the semi-minor axis (the shorter). Enter both and this calculator returns the area, the circumference, the eccentricity, and the position of the two foci, drawn true to scale.

The parts of an ellipse

Semi-major axis — half the length of the longest diameter, through the centre.
Semi-minor axis — half the length of the shortest diameter, at right angles to the major.
Foci — two fixed points on the major axis. The sum of the distances from any point on the ellipse to the two foci is constant.
Eccentricity — a number from 0 to 1 describing how stretched the ellipse is. Zero is a perfect circle; values near 1 are long and thin.

Worked example

Take an ellipse with semi-major axis a = 5 and semi-minor axis b = 3.

Step 1 — area

A = π a b = π × 5 × 3 = 47.12

Step 2 — focal distance from the centre

c = √(a² − b²) = √(25 − 9) = √16 = 4

Step 3 — eccentricity

e = c / a = 4 / 5 = 0.8

Step 4 — circumference (Ramanujan's approximation)

C ≈ π(a+b)[1 + 3h / (10 + √(4 − 3h))], h = ((a−b)/(a+b))² ≈ 25.53

Result

Area 47.12, circumference ≈ 25.53, focal distance 4, eccentricity 0.8. The foci sit 4 units either side of the centre along the major axis — and because a and b form a 3-4-5 relationship, c comes out as a whole number.

Formulas this calculator uses

Area · A = π a b

Circumference · π(a+b)[1 + 3h/(10+√(4−3h))], h = ((a−b)/(a+b))²

Focal distance · c = √(a² − b²)

Eccentricity · e = c / a = √(1 − b²/a²)

Here a is taken as the longer semi-axis when computing eccentricity and the foci, so the foci always fall on the major axis whichever box is larger. The circumference uses Ramanujan's second approximation, accurate to about one part in ten million even for very stretched ellipses.

Frequently asked questions

How do you find the area of an ellipse?

Multiply pi by the two semi-axes: A = π a b. It's the direct generalisation of a circle's πr² — a circle is just an ellipse where a and b are equal. An ellipse with semi-axes 5 and 3 has an area of about 47.12.

How do you find the circumference of an ellipse?

There's no exact elementary formula — the true circumference is an elliptic integral. In practice we use a very close approximation. This calculator uses Ramanujan's formula, which is accurate to many decimal places for any realistic ellipse.

What is the eccentricity of an ellipse?

Eccentricity measures how far an ellipse departs from a circle, on a scale from 0 to 1. It's e = √(1 − b²/a²), where a is the larger semi-axis. A value of 0 is a perfect circle; values approaching 1 are long and thin.

What are the foci of an ellipse?

The foci are two fixed points on the major axis, each a distance c = √(a² − b²) from the centre. The defining property of an ellipse is that the distances from any point on the curve to the two foci always add up to the same total.

What's the difference between the major and minor axis?

The major axis is the longest diameter of the ellipse; the minor axis is the shortest, at right angles to it. "Semi" means half, so the semi-major and semi-minor axes are the distances from the centre to the edge along each direction.

Is a circle an ellipse?

Yes — a circle is the special case of an ellipse where the two semi-axes are equal. Its eccentricity is zero, both foci sit together at the centre, and the circumference reduces to 2πr.