Circumference of an oval

Answer 1

There is no simple answer: in fact the search for the answer led to the study of elliptical integrals.

C = 2*pi*a*{1 - [1/2]^2*e^2 + [(1*3)/(2*4)]^2*e^4/3 - [(1*3*5)/(2*4*6)]^2*e^6/5 + ... }

where

a is the semi-major axis,

b is the semi-minor axis, and

e is the eccentricity = sqrt{(a^2 - b^2)/a^2}.

The above infinite series converges very slowly.

An approximation, suggested by Ramanujan, is

C = pi*{3(a + b) - sqrt[(3a + b)*(a +3b)]} = pi*{3(a + b) - sqrt[10ab + 3(a^2 + b^2)]}