Formula for the Circumference of an Ellipse

Date: 6/27/96 at 14:1:21
From: mike reese
Subject: Circumference of an Ellipse

Please divulge the correct formula or equation for solving the 
circumference of an ellipse.

Date: 6/28/96 at 14:34:15
From: Doctor Mike
Subject: Re: Circumference of an Ellipse

Hello,

Nice to hear from another Mike.  This is an interesting question, 
and I did not know the answer, so I looked it up on our web site (see 
below).  Turns out it is a VERY interesting question with a lot of 
history.  There is a simple formula for the area of an ellipse, but 
not for the circumference (or perimeter).  Here's what I found out.
  
I started out by picking the SEARCH DR. MATH ARCHIVES link on the Web 
site and entered ELLIPSE as the search word.  I got a lot of things to 
look at, which led me to further links, and eventually found my way to 
a document with the net URL address:

  http://www.astro.virginia.edu/~eww6n/math/math.html   

This is literally a Treasure Trove of math information, and I looked 
up the Ellipse subject in this alphabetical arrangement.
  
If the ellipse has semi-radius values A and B, then the area is  
pi(A*B)  which is a nice generalization of the pi(R^2)
formula for the circle.  The only available formulas for the
perimeter involve Elliptic Integrals, which doesn't do you much
good.  There is an approximation formula by Ramanujan: 

       pi( (3A+3B) - sqrt((A+3B)*(B+3A)) )

which is about as good as you are going to get. I invite you
to look around at these Internet sites.  Enjoy.
  
I hope this helps.

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/