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Volume of ellipsoid

Date: 05/22/97 at 22:51:42
From: Anonymous
Subject: Calculation of the volume of ellipse

Dear Dr. Math,

I have forgotten how to calculate the volume of an ellipse. I guess 
the volume of ellipse with radius of a and b should be (2/3*pi*a^2*b) 
where a is shorter radius and be is longer radius.

Would you teach me the soultion and tell me where I can find 
information in internet?

With my best regards,
Wonil Chung

Date: 05/23/97 at 06:17:27
From: Doctor Mitteldorf
Subject: Re: Calculation of the volume of ellipse

Dear Wonil Chung,
Language is making it hard for us.  An ellipse is a plane figure, in 2
dimensions.  Volume is 3-dimensional.  I think you want to ask about 
the volume of an ellipsoid.  An ellipsoid is a 3-D figure, like a 
sphere, but it can have 3 different radii in 3 directions. Maybe you 
are thinking of an "oblate ellipsoid of revolution" where twoof these 
are the same.  This is a watermelon shape, or a cigar shape.
The formula for the volume of a sphere is 4/3 pi r^3.  You can scale 
each of the 3 dimensions separately, and generalize this to the volume 
of an ellipsoid with semi-axes a, b, and c.  The answer is just 
4/3 pi a*b*c.  If two of them are the same, the volume is 4/3 pi a^2*b
Any math handbook has these formulas.  

Have you studied calculus?  If you set up the integrals for a sphere 
and for an ellipsoid, you can easily show the relation between the two
volume formulas.  

You might also just think about it and see if you can convince 
yourself that it is true that the formula for the volume of an 
ellipsoid is just the same as the formula for a sphere, but with each 
axis "stretched" appropriately.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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