Volume of a Frustum of a Pyramid

Date: 06/03/2002 at 13:36:25
From: Zizza
Subject: the frustum of an pyramid

Hi Dr. Math.

I don't know how to prove that the formula 

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

is correct for a frustum of a pyramid.

Date: 06/03/2002 at 22:40:14
From: Doctor Peterson
Subject: Re: the frustum of an pyramid

Hi, Zizza.

Here is one way to derive the formula:

If we reconstruct the entire pyramid, the top part, with base area 
B2, is similar to the whole pyramid, with base area B1. Their heights 
must therefore be in the ratio sqrt(B1):sqrt(B2). The height of the 
whole pyramid is therefore sqrt(B1)/(sqrt(B1)-sqrt(B2)) h, and its 
volume is

    Vwhole = 1/3 B1 * sqrt(B1)/(sqrt(B1)-sqrt(B2)) h

while the volume of the removed top part is

    Vremoved = 1/3 B2 * sqrt(B2)/(sqrt(B1)-sqrt(B2)) h

Subtracting, we get

    Vfrustum = Vwhole - Vremoved

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

             = h/3 * [sqrt(B1)^3 - sqrt(B2)^3]/[sqrt(B1)-sqrt(B2)]

But the difference of cubes can be factored:

    a^3 - b^3 = (a - b)(a^2 + ab + b^2)

so we get

    Vfrustum = h/3 * (B1 + sqrt(B1*B2) + B2)

which is just what we wanted.

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