Ellipse Calculator

How it works

An ellipse is a stretched circle. Its longest diameter is the major axis and its shortest the minor axis; half of each, measured from the centre, are the semi-axes a and b. The area is π × a × b, exactly as a circle's is π × r × r, because an ellipse is a circle scaled by a different factor in each direction.

The perimeter has no simple exact formula; it is an elliptic integral. The calculator uses Ramanujan's second approximation from 1914, which is the standard practical formula. For ordinary shapes it is exact to more decimal places than are shown; even for a very long, thin ellipse it is within 0.04 %.

Every ellipse has two foci on the major axis, each a distance c = √(a² − b²) from the centre. The distances from any point on the edge to the two foci always add up to 2a, which is why you can draw an ellipse with two pins and a loop of string. The eccentricity e = c ÷ a measures how stretched the ellipse is: 0 is a circle, and values near 1 are long and thin.

Formula

area      = π × a × b
h         = ((a − b) ÷ (a + b))²
perimeter ≈ π (a + b) (1 + 3h ÷ (10 + √(4 − 3h)))     Ramanujan, 1914
c         = √(a² − b²)      e = c ÷ a

Example

An ellipse with semi-axes a = 5 ft and b = 3 ft has an area of π × 5 × 3 = 15π ≈ 47.12 ft². With h = (2 ÷ 8)² = 0.0625, Ramanujan's formula gives a perimeter of π × 8 × (1 + 0.1875 ÷ (10 + √3.8125)) ≈ 25.53 ft. The foci are c = √(25 − 9) = 4 ft from the centre, and the eccentricity is 4 ÷ 5 = 0.8.

A 2 : 1 ellipse with a = 2 and b = 1 has area 2π ≈ 6.2832 and perimeter 9.6884, which matches the exact value 8 × E(60°) = 9.688448 from tables of the complete elliptic integral to all the digits shown.

Assumptions and limitations

• The perimeter is Ramanujan's approximation, not an exact value. It is accurate to better than 1 part in 50,000 for any ellipse up to 10 : 1 and never off by more than about 0.04 %, even in the degenerate limit.
• All lengths are in the unit you chose. The calculator labels the results with it and does not convert between units.
• The figure is a complete ellipse. A partial ellipse such as an elliptical arch is not handled here.
• Results are rounded for display only.

Frequently asked questions