Volume of a Pyramid

Date: 08/30/98 at 07:45:10
From: Terence Tham
Subject: Volume of pyramids

How come the volume of a pyramid is equal to 1/3 the volume of a prism 
of the same base area and height? Can you explain it to me? 

Thanks.

Date: 08/31/98 at 12:03:31
From: Doctor Peterson
Subject: Re: Volume of pyramids

Hi, Terence. A lot of students wonder about this, and we seldom bother 
to give a proof, because it generally is beyond the abilities of 
students at the level we teach the formula. There is an explanation 
using infiinite series in our Dr. Math archives at

   Volume of a Cone or Pyramid
   http://mathforum.org/dr.math/problems/swan3.30.98.html   

This is rather brief, so if you need help understanding it, write back!

You may also be interested in seeing how Euclid proved it (complete 
with a picture, if your browser supports Java):

   Book XII, Proposition 7, Euclid's Elements - David Joyce
   http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII7.html   

This shows a way to divide a triangular prism into three pyramids of 
equal volume; a little more work is needed to complete the proof for 
all pyramids and for cones, which Euclid does next.

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

Date: 09/01/98 at 06:56:11
From: Terence
Subject: Volume of pyramids

I do not understand the information on the Web site you sent me. Can 
you further explain it to me? Thanks.

Date: 09/01/98 at 12:04:49
From: Doctor Peterson
Subject: Re: Volume of pyramids

Hi, Terence. I was expecting to hear back from you. This proof is a 
little complicated, which is why we don't generally bother to teach 
it. Most students are satisfied either to trust mathematicians, or to 
try it out by measuring the volume of a pyramid and seeing that the 
formula works. It's good that you're not satisfied with that, and want 
to know the reasons behind what you learn.

The first page I referred you to, at    

 http://mathforum.org/dr.math/problems/swan3.30.98.html   

gives this calculation for the volume of a square pyramid:

   V = a^2*b*[1^2 + 2^2 + 3^2 + ... + n^2]
     = a^2*b*n*(n + 1)*(2*n + 1)/6  (which can be proved by induction)
     = a^2*b*n^3*(1/3 + 1/[2*n] + 1/[6*n^2])

Here's what it means:

Suppose we try to build a square pyramid out of flat square boxes 
(whose volume we can calculate as L x W x H). It won't be perfect, but 
will look like a "step pyramid":

                    *-* ----------------------------------
                  **-*|----*                            ^
                */ | |*   /|----*                       |
              *//  *-*   / *   /|----*                  |
            *//*--------* /   / *   /|----*             |
           /// |        |/   / /   / *   /|b            H
          ///  *--------*   / /   / /   / *             |
         //*---------------* /   / /   / /              |
        // |               |/   / /  a/ /               |
       //  *---------------*   / /   / /                v
      /*----------------------* /   / /B -----------------
     / |                      |/   / /
    /  *----------------------*   / /
   *-----------------------------* /
   |              a              |/
   *-----------------------------*
                  B

Let's build it so that the base of each horizontal slab is a cross-
section of the pyramid we are trying to measure, so that this step 
pyramid is larger than the real pyramid. Say the base is B by B, so 
its area is B^2, and the height is H. If we have divided the height 
into N slabs, then the height of each slab is H/N, and the width of 
the K'th slab from the top is B*K/N. (Do you see why? The top slab is 
B/N, the next is 2B/N, ..., the bottom one is B*N/N = B.)

(Note that his a is my B, and his b is my H/N.)

Then the volume of the whole thing is the sum of the volumes of the 
slabs:

   (B*1/N)^2 * H/N + (B*2/N)^2 * H/N + ... + (B*N/N)^2 * H/N

and we can factor out (B/N)^2 * H/N from all the terms to get:

   (B/N)^2 * H/N * (1^2 + 2^2 + ...+ N^2)

There is a formula for the sum of squares, which is:

   1^2 + 2^2 + ...+ N^2 = N*(N + 1)*(2N + 1)/6

This can be proved by induction; that is, by showing that it is true 
for N = 1, and that if it is true for N, it is also true for N+1. Once 
that is proved, it must be true for all N. Here is a quick proof:

   For N = 1, 1^2 = 1*2*3/6 is true.
   If true for N, then
   1^2 + ... + N^2 + (N+1)^2 = N*(N + 1)*(2N + 1)/6 + (N+1)^2
                             = (N+1) * [N*(2N+1)/6 + (N+1)]
                             = (N+1) * [(2N^2 + N) + (6N + 6)] / 6
                             = (N+1) * [2N^2 + 7N + 6] / 6
                             = (N+1) * (N+2) * (2N+3) / 6
                             = (N+1) * ((N+1) + 1) * (2*(N+1) + 1) / 6
   so the formula is true for N+1.

If we use this formula, we get the volume of the step pyramid as:

   V = (B/N)^2 * H/N * N*(N + 1)*(2N + 1)/6

       B^2 * H * N * (N+1) * (2N+1)
     = ----------------------------
           N^3 * 6

       B^2 * H   2N^3 + 3N^2 + N
     = ------- * ---------------
          6             N^3

       B^2 * H         3     1
     = ------- * (2 + --- + ---)
          6            N    N^2

Now, if we use smaller and smaller steps (bigger and bigger N), our 
step pyramid will get closer and closer to the actual pyramid. But if 
N is very large, 3/N and 1/N^2 are much smaller than 2, so we can 
ignore them. (They represent the extra part of the steps that we have 
to cut off to get the pyramid.) Then the formula becomes:

   V = B^2 * H / 3

That's what we were looking for!

Now, Euclid's proof in:

 http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII7.html    

uses no algebra, but more geometric reasoning. The main part of the 
proof is to show that a triangular prism can be divided into three 
equal pyramids, so that the volume of one of the pyramids is one third 
of the volume of the prism that contains it. Given that, you can 
easily extend the formula to prisms with any shape base, since the 
base of any pyramid can be divided into triangles. A few propositions 
later, he extends the formula to cones.

Here's how he divides the prism:

               F
                +
            /   |\
    E   /       | \D
    +--------------+
    |           |  |
    |           |  |
    |           |  |
    |           |  |
    |          C|  |
    |           +  |
    |       /    \ |
    |   /         \|
    +--------------+
    B              A

is equal to the sum of these three pyramids:
                                                         F
                                                          +
                                                      /   |\
                   D     E              D     E   /       | \D
                   *     *--------------*     *--------------*
                 //|     | \           /        \         | /
               /  /|     |   \       / /          \       | /
              /  / |     |     \   /  /             \     |/
            /    / |     |       /    /               \   |/
          /    C/  |     |      /  \ /                  \ |
         /      *  |     |    /      *                    *
       /    /    \ |     |  /    /   C                    C
     /  /         \|     |/  /
    *--------------+     *
    B              A     B

Each can be shown to be equal to another because it has the same base 
area and the same altitude, which Euclid had shown earlier to imply 
the same volume, even without knowing the actual formula yet.

So that gives you two ways to prove the formula. There may still be 
some hard parts here for you, but work at it and it should become 
clear.

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

Date: 09/03/98 at 04:16:42
From: Tham Terence
Subject: Re: Volume of pyramids

Hi. Can you send me simpler proofs which I can understand? Can the 
hypoteneuse be shorter than the adjacent side or opposite sides? I am 
learning the Pythagorean Theorem in school.

Date: 09/03/98 at 08:52:55
From: Doctor Peterson
Subject: Re: Volume of pyramids

Hi again. As I explained, we usually don't try to prove the volume 
formulas for cones, spheres, and so on precisely because they are 
beyond most students of your age. Hold on to that first proof until 
you have had enough algebra, and the second until you have had enough 
geometry, then study them and you should be convinced. I think each of 
them is about as simple as I can make it. I consider the second proof 
(Euclid's) to be the best, because it is something you can see. If you 
spend enough time with it (maybe even making models of a prism split 
into pyramids) you should be able to follow it enough to believe it.

If you think about the Pythagorean theorem:

   a^2 + b^2 = c^2

you can see that the hypotenuse has to be bigger than either side. For 
example, c must be bigger than a because you are adding a positive 
number (b^2) to a^2 to get c^2, so c^2 must be bigger than a^2 and c 
must be bigger than a.

I'm glad you want to ask questions beyond your own level. Keep 
thinking and working at math, and you will find yourself understanding 
more and more.

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