Area, Circumference of an Ellipse

Date: 7/29/96 at 3:33:35
From: Anonymous
Subject: Area, Circumference of an Ellipse

Hello,

How do I calculate the area and circumference of a given ellipse?

Thanks,
    Magnus

Date: 7/29/96 at 13:24:49
From: Doctor Jerry
Subject: Re: Area, Circumference of an Ellipse

I'm not certain of the level at which you are asking your question.

First, if the ellipse is described by the equation 
x^2/a^2+y^2/b^2=1, 
then the area enclosed by the ellipse is pi*a*b.

There is no similar formula for the "circumference" or length of an 
ellipse.

If you know calculus, here are two integrals for the area enclosed by 
the ellipse and its length.

For area, solve the above equation for y to obtain
y = (b/a)*sqrt(a^2-x^2) for the top half.  The area is
A = 4*(integral from 0 to a of (b/a)*sqrt(a^2-x^2) dx)

For the length it is simpler to use the parametric description of the 
ellipse.  It is x=a cos(t), y=b sin(t).  The formula for the length is
4*(integral from 0 to pi/2 of sqrt(a^2 sin^2 t+b^2 cos^2 t) dt).

This integral is a so-called elliptic integral and must be done 
numerically or looked-up in a table.

I hope this answer is useful.  If not, please ask again.

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