Deriving the Volume of a Frustum

Date: 05/06/98 at 09:36:51
From: Mike Taylor
Subject: Physics - volumes

I need to find the equation for the volume of a frustum of a cone. I 
have searched everywhere for the answer and have not been able to find 
it. Thank you.

Date: 05/06/98 at 12:19:40
From: Doctor Tom
Subject: Re: Physics - volumes

Hi Mike,

A frustum of a cone is the piece left over after you've cut a smaller
cone off the top of a larger cone in such a way that the cut is 
parallel to the base of the original.

So if you know how to find the volume of a cone, you can take the
volume of the original cone and subtract the volume of the piece 
you cut off.

The formula, of course, depends on what values you're given. I'll
assume you know the radius of the base of the larger cone (let's call 
that "R"), the radius at the top where you cut off the smaller one 
(let's call that "r"), and the height of the frustum - the distance 
between the flat faces (call that "h").

The side view looks like this:

         ^
        / \
       /   \
      /_____\      _
     / r     \     |
    /         \    | h
   /___________\   |
     R

The frustum is just the lower part, but I've shown the entire
original cone above. Let's let "H" be the height of the original cone, 
so by similar triangles:

    H/R = (H-h)/r

So rH = R(H-h), or rH - RH = -Rh, or H(R-r) = Rh, or H = Rh/(R-r)

The volume of the original cone is (1/3)*pi*R^2*H, and the volume of 
the chunk that's cut off is (1/3)*pi*r^2*(H-h).

The volume of the frustum is (1/3)*pi*(R^2*H - r^2*(H-h)), and you 
know the value of H: it's H = Rh/(R-r).  Plug that in and simplify it,
and you're in business.

For more information on cones and their frustums, see the Dr. Math 
FAQ:

  http://mathforum.org/dr.math/faq/formulas/faq.cone.html   

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

Date: 05/07/98 at 09:08:34
From: Mike Taylor
Subject: Re: Physics - volumes

Thank you very much for your help on the frustum problem. I really
appreciate it and feel very enlightened.

Mike