The two dimensional version of this is to find the area of a trapezoid, so let's start with finding the area of a trapezoid using geometric series and then extend the result to 3 dimensions. The procedure used to find the area of a trapezoid will be a bit roundabout, but it will be very easy to extend the results to 3 dimensions.

The strategy for finding the area of the trapezoid is to build a triangle by layering copies of the trapezoid as show in the illustration. We will use the geometric series to express the area T of the triangle in terms of the base trapezoid . We will then find another geometric series to find the area of the triangle. We will then equate the two expressions for the area of the triangle to allow us to solve for the area of the trapezoid.

The trapezoid whose area A we want to find is ABCD given the lengths of the two bases b

The next thing we want to do is to find the area of the nth trapezoid in terms of A, r and n. Since the scaling factor is r

T = A(1 + r

T = (1 - r

Next, let's find the area of the triangle in terms of b

The area of the triangle is therefore T = 1/2 b

We could at this point substitute b

A/(1 - r

A = 1/2 b

(1 - r

Using the simplification, we get

A = 1/2 b

Substituting b

It is now relatively easy to find the volume of a square truncated pyramid using the technique we used for trapezoids.

We have b

The height of the pyramid is h/(1 - r) , Using this in the formula for the volume of a pyramid, P = 1/3 b

Equating the two equations for P:

V/ (1 - r

V = 1/3 b

Recognizing (1 - r

V = 1/3 b

Do you see a pattern in going from two dimensions to three dimensions? What would you expect the formula to be in four dimensions?

Solution Without
Infinite Series

Here is an alternative way to get the two
equations for the
volume of the pyramid. It avoids using the two infinte
geometric
serices to derive the two equations. We still make use of
finite
geometric series when combining thte two equations. The basic idea is actually pretty simple. The small pyramid above the frustum is similar to the whole pyramid, with scaling factor r = b

Let V' = volume of small pyramid lying on top of frustum.

P = V + V'

V'/(V+V') = r

1 - V'(V + V') = 1 - r

(V+V')/(V+V') - V'/(V+V') = 1 - r

V/(V+V') = 1 - r

V/P = 1 - r

Let h' = height of small pyramid lying on top of the frustum. The height of the whole pyramid is h + h'.

h'/(h + h') = r.

1 - h'/(h + h') = 1 - r

(h + h')/(h + h') - h'/(h+h') = 1 - r

h/(h + h') = 1 - r

h + h' = h/(1 - r)

We can now plug this value into the second equation for the volume P of the pyramid.

More General Case

The formula given for the frustum only covers
the case of square pyramids. Suppose instead we are given a
base with any general shape having area B