Calculations at an ellipse. Enter the two semi axes and choose the number of decimal places. Then click Calculate. The semi-major axis is the distance between center and one point on the ellipse furthest away from this. The semi-minor axis is the distance between center and one point on the ellipse closest to this. They are upright to each other. The linear eccentricity is the distance of the focal points to the center. The circumference is calculated with an approximation formula (second approximation of Srinivasa Ramanujan), which is very exact up to ε<0.9 and has a maximal error of 0.04% at ε=1. An exact calculation can be done with elliptic integrals (Jacobi integrals), whose values can be taken from tables.
c_{B} is an about 7 times better estimation, expending the second approximation of Ramanujan, by Jürgen Beck. This is based on an astroid approximation and is not verified yet.

Ellipse shape:

Formulas:
e = √a² - b²
ε = e / a
A = π * a * b

Circumference after the second approximation of Ramanujan:
c ≈ π * (a+b) * [ 1 + 3λ² / (10+√4-3λ² ) ]
λ = ( a - b ) / ( a + b )

Circumference after the second approximation of Ramanujan expanded by Jürgen Beck:
c_{B} = u + 0.00160934997662698*(1-x^{0.8})^{21}
x = {2/[(a/b)^{bex}+1]}^{(1/bex)}
bex = ln(2) / ln(π/2)

The number 0.00160934997662698 is the difference between the exact value for the circumference of the ellipse, after a numeric integration according the calculated Taylor series and the value after the second approximation of Ramanujan, for a=1 and b=0.

pi:
π = 3.141592653589793...

Semi axes, linear eccentricity and circumference have the same unit (e.g. meter), the area has this unit squared (e.g. square meter). Numerical eccentricity is dimensionless.