How this calculator works
An ellipse is a stretched circle with two semi-axes: a is the longer semi-major axis and b is the shorter semi-minor axis. Its area is exactly πab, so knowing the area and one axis fixes the other. The eccentricity measures how far the ellipse is from a circle, ranging from 0 for a circle up to just below 1 for a very elongated ellipse. The linear eccentricity is the distance from the centre to either focus. Unlike the area, the perimeter has no elementary closed form; this calculator uses Ramanujan's second approximation, which is accurate to better than one part in a million for nearly all ellipses.
Formula
Area = πab; c = √(a² - b²); eccentricity e = c/a; b = A/(πa); a = A/(πb); b = a√(1 - e²); a = b/√(1 - e²); b = √(a² - c²); a = √(b² + c²). Perimeter ≈ π[3(a + b) - √((3a + b)(a + 3b))].
Worked example
An ellipse with semi-axes 5 m and 3 m has area π × 5 × 3 ≈ 47.1239 m², linear eccentricity √(25 - 9) = 4 m, eccentricity 0.8 and perimeter about 25.5270 m.
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Frequently asked questions
Is the perimeter exact?
No. The exact perimeter is an elliptic integral. Ramanujan's approximation used here is extremely accurate for ordinary ellipses, but it is still an approximation.
What is the difference between eccentricity and linear eccentricity?
Eccentricity is a dimensionless number between 0 and 1 that describes the shape. Linear eccentricity is an actual length, the distance from the centre to a focus.
Can the two axes be equal?
Yes. Equal axes describe a circle. The eccentricity is then zero, and the linear eccentricity is zero as well.
Why is there no semi-major axis smaller than the semi-minor axis?
The names mean the longer and shorter half-axes. If your numbers are reversed, swap them so that a is the larger value.