Volume of a Frustum-Like Structure

Date: 05/12/2000 at 13:13:25
From: Alison Cunningham
Subject: Volume of a Frustum

I have a pyramid-like structure with a rectangular base and 
rectangular top, i.e. the top of a rectangular pyramid has been 
removed. I have tried using three different methods to calculate the 
volume. I was told it's called a frustum.

Which one do you suggest?

Top:    73 by 37
Bottom: 46 by 10.5
Angle:  18 degrees
Height: 4.6

Date: 05/12/2000 at 20:56:28
From: Doctor Peterson
Subject: Re: Volume of a Frustum

Hi, Alison.

I don't know what three methods you have, but our Formula FAQ has a 
formula:

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

(scroll down to Frustum of a Pyramid):

     V = h(B1+B2+sqrt[B1*B2])/3

Unfortunately, your shape is not really a frustum of a pyramid, 
because the top and bottom are not similar, so if you continued the 
edges up to the point of the presumed pyramid, they would not meet. 
So that formula does not work.

I used calculus, though it probably wasn't entirely necessary, to find 
a formula for this sort of shape where two rectangles are joined by 
straight edges, without having to be a frustum, and I got this:

                  a2
              +-------+
           b2/       / \---------
            +-------+   \      |
           /        |    \     |
          /.........|.... +    | h
         /          |    /     |
        /           |   /      |
       /            |  /b1 ------
      /             | /
     /              |/
    +---------------+
           a1

     V = [a1*b1 + a2*b2 + (a1*b2 + a2*b1)/2] * h/3

Here the bottom is a1 x b1, and the top is a2 x b2, with the a's 
parallel and the b's parallel. Notice that a1*b1 = B1 and a2*b2 = B2 
in the frustum formula; but the third term "averages" the two in a 
different way, which works out to be the same if a1/b1 = a2/b2. Try 
both formulas for your shape, and you'll find they give different 
results.

If you are supposed to be able to figure this out for yourself, I'll 
try to see how you could do it; otherwise, just use this formula, 
which I don't recall ever seeing.

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