Here’s my best approximation of the perimeter of an ellipse.
One thing you might notice is most of the approximations Matt presents get progressively worse as 
But that’s sort of the nature of approximations, when what you’re trying to approximate increases without bound: If your approximation doesn’t capture the exact behavior (and if it does, it’s not really an approximation, is it?) then it’ll necessarily diverge from the correct value by more and more. Really the best you can hope for is to get a good approximation within some bounds, knowing it’ll go increasingly bad once you’re far enough outside those bounds.
So I thought I’d see if I could get the divergence to occur much further out than
Ramanujan’s approximation is:
I cracked open a spreadsheet and found that if 
where 
This formula doesn’t work for 
And here’s a Python function to calculate the corrected approximation (for a/b > 3, otherwise it returns Ramanujan’s approximation):
#!/usr/bin/python
from sys import argv
from math import pi
def ellper (a, b):
q = 0.0004714
r = 1.6156
pram = pi*(3*(a+b)-((3*a+b)*(a+3*b))**0.5)
return pram + pi*(q*b*(a/b-3)**r) if a/b >= 3 else pram
a = float(argv[1])
b = float(argv[2])
print ellper (a, b)
