Circumference of an Ellipse (Euler)

Circumference of an Ellipse (Gauss-Kummer)

eccentricity ε =

semimajor axis a =

number of terms n =


semiminor axis b =

circumference S =

In a paper, published in 1738, Euler found a series for the arc length of an ellipse:

Circumference S = 2π*a*(1 + c1 ε2 + c2 ε4 + c3 ε6 + c4 ε8 + ...)

coefficients ci
terms ci ε2i

The convergence of the series is very slow when the eccentricity ε is almost 1:

n=10 S=4.423712404139952
n=20 S=4.412502156808594
n=30 S=4.411186402526338
n=50 S=4.410902681269847
n=85 S=4.410886764405142

There are some more series having better convergence properties.

Web Links

How Euler Did It - Arc length of an ellipse

Circumference of an Ellipse

Specimen de constructione aequationum differentialium sine indeterminatarum separatione

(c) 2006-2016 J. Giesen