Volume of a Cone or Pyramid

Date: 03/30/98 at 16:04:10
From: Laura Lee Swan
Subject: Volume of cone/pyramid

Hi there,

I have been happily entertained for the last 40 minutes searching your 
web site for proof that the volume of cone/pyramid = (1/3)b*h. I found 
something, but it mentioned Riemann sums, so I stopped. Can you 
explain the rest?

I've been searching for an answer for weeks and can't get my hands on 
the right text book or person. My superstar 8th grader wants to know 
why, and as a permanent sub (I am not a permanent math teacher who 
ought to know how to explain this), I'm stumped. I'm thrilled that 
he's asking, so I want to be able to give him an answer (no matter how 
difficult).

Thanks.

Date: 03/30/98 at 22:55:45
From: Doctor Rob
Subject: Re: Volume of cone/pyramid

Imagine a step-pyramid made up of small square prisms with dimensions
a-by-a-by-b, arranged in square layers. If the layers are k-by-k for
k = 1, 2, 3, ..., n, then the total volume of the step-pyramid is:

   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]).

Now, the height of the step-pyramid is h = b*n, and the area of the 
base is:

   B = a^2*n^2, 

so the volume is:

   V = h*B*(1/3 + 1/[2*n] + 1/n^2).

Now let n -> infinity, so the steps get smaller and smaller, and the
volume of the step-pyramid approaches the volume of the inscribed 
pyramid. Then the right-hand side approaches h*B/3, as the volume V 
approaches the volume of the pyramid.

This same proof works for a pyramid with any polygonal base. For a 
cone, take a pyramid with a regular m-gon for a base, and let 
m -> infinity.

The base approaches a circle, and the pyramid approaches a cone, and 
the formula V = h*B/3 still holds in the limit.

There is another proof, which cuts a rectangular solid into six pieces 
of equal volume, each of which is a triangular pyramid (or irregular
tetrahedron). This proves that the volume of each is 1/6th the volume 
of the starting rectangular solid, and each has base of area 1/2 the 
area of one of the faces of the solid, and height equal to the 
dimension of the solid perpendicular to that face.

-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 01/14/99 at 20:34:06
From: Sandy Deering
Subject: Volume of Cone

Dear Dr. Math,

I read this letter on the Volume of a Cone or Pyramid but I still 
didn't find my answer. My question is: What does the formula for the 
volume of a cone (1/3 * pi * r^2 * h) really mean? Why do we use that 
formula?

Date: 01/15/99 at 00:57:39
From: Doctor Pat
Subject: Re: Volume of Cone

Sandy,

The volume of a cone may be thought of as a special case of the volume 
of a cylinder. The volume of a prism-like object is equal to its base 
area times its height. That is true for any solid for which each cross 
section is identical at any height. A box and a cylinder are examples. 
This is known as Cavalerri's principal. That is where the Pi*r*r comes 
into the problem. The base area of a cone and cylinder is a circle. 

If the base on the top is reduced to a similar shape, but smaller 
size, the volume decreases. It doesn't matter what shape the top and 
bottom are, this is true for all of them. If we reduce the similar 
shape at the top to a single point, we make a pyramid or cone. The 
volume of this new figure is 1/3 the volume of the prism-like form.  
We need calculus to prove this rigorously. To see how, take a look at:

  http://mathforum.org/dr.math/problems/swimduck5.29.96.html   

But to just convince yourself only requires some appropriate cylinders 
and cones with the same base. Ask your teacher if your school has a set 
of volume containers in these shapes, and then pour rice from the cone 
to the cylinder. Three conefuls should fill it right up.    

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