Volume of a Prismatoid

Date: 10/21/2008 at 10:05:49
From: Ravi
Subject: Volume of circular end pyramid

How can I calculate the volume of a pyramid for which the bottom end
is rectangular and the top end is circular?  I'm totally confused.

Date: 10/21/2008 at 10:49:23
From: Doctor Peterson
Subject: Re: Volume of circular end pyramid

Hi, Ravi.

This is not really a pyramid, but I picture it as a sheet-metal 
transition from a round pipe to a rectangular duct, which is formed 
by bending a series of triangles with one base in one end of the 
figure, and the vertex in the other.  Is that right?

The volume can be calculated using the formula for a prismatoid, 
which is the same kind of shape except that the top is a polygon 
rather than a circle:

  Prismatoid
    http://mathworld.wolfram.com/Prismatoid.html 

The formula is

  V = h/6 (A1 + 4M + A2)

where A1 and A2 are the top and bottom areas, and M is the midsection 
area.  In our case, if

  r = radius of circle
  L = length of rectangle
  W = width of rectangle

it turns out that

  A1 = pi r^2
  A2 = LW
  M  = (L/2 + r)(W/2 + r) - r^2 + pi r^2/4

and so

  V = h/3 (pi r^2 + LW + Lr + Wr)

That is such a nice formula, I imagine I am not the first to write it!

See if this works for you.  If you have any further questions, feel 
free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/