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Volume of an Elliptical Frustum

Date: 12/01/1999 at 07:46:49
From: Alan Macleod
Subject: An elliptical frustum

Dear Doc,

I'm still confused! Please help. From

   Deriving the Volume of a Frustum   

the volume of a frustum is

     V = (1/3)*pi*(R^2*H - r^2*(H-h))


     H = Rh/(R-r)

From other articles:

The volume of a cone is

     V = (1/3)*pi*R^2*h

and the volume of an elliptical cone is

     V = (1/3)*pi*a*b*h

Can I therefore assume that the volume of a elliptical frustum is:

     V = (1/3)*pi*[(a*b)*H - (c*d)*(H-h)]


     H = ---------------------
         sqrt(a*b) - sqrt(c*d)


     a = semi-major axis of base
     b = semi-minor axis of base
     c = semi-major axis of top
     d = semi-minor axis of top
If not, then can you give me the correct formulas?

Date: 12/01/1999 at 10:02:08
From: Doctor Rob
Subject: Re: An elliptical frustum

Thanks for writing to Ask Dr. Math, Alan.

Yes, the volume of an elliptical frustum is

     V = (1/3)*pi*[(a*b)*H - (c*d)*(H-h)]

The last part is also correct, but you can avoid the square roots if 
you realize that a/c = b/d = H/(H-h). Then you can use instead

     H = a*h/(a-c) = b*h/(b-d)

- Doctor Rob, The Math Forum   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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