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### 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/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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