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Area, Circumference of an Ellipse

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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
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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/
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Associated Topics:
College Calculus
College Euclidean Geometry
High School Calculus
High School Euclidean/Plane Geometry

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