Volume equations for a sphere and pyramid

Date: 6/10/96 at 18:59:48
From: Ron C. Maggiano
Subject: How do you get the volume equations for a sphere and pyramid?

Hey Dr. Math,

We just learned the formulas for volume of a sphere (one third of four 
times the side to the cube), and volume of a pyramid (one third of base 
times the height,) but our geometry teacher wouldn't tell me how to get 
them. Can you help?

Thanks!

Date: 7/1/96 at 17:0:16
From: Doctor Paul
Subject: Re: How do you get the volume equations for a sphere and 
pyramid?

The Volume of a Sphere is:
                  
  4 * Pi * (radius)^3
  -------------------
           3


The Volume of a pyramid is:

(Area of Base) * (height)
 ------------------------
           3


I think the reason your teacher wouldn't tell you where these formulas 
come from is that you probably wouldn't understand. They are derived 
from Calculus.  Here's the basic idea. 

Let's do the volume of a sphere. It's a little more simple than a 
pyramid, although the ideas are similar. I'm going to tell you how to 
get the volume of a semi-sphere (or half of a sphere).  I think it's 
obvious that once you get half of it, you can double your answer and 
get the volume of the entire sphere.  

Let's work with the top half of the sphere (it would look like a 
bird cage).  Note that at each point the radius is a different length.  
At the top of the bird cage the radius is zero, and at the bottom it 
is some pre-determined number.  Just note that it is not constant.
Now... in your mind, make hundreds of little horizontal slices that 
cut across the sphere.  Horizontal is NOT up and down.  It is from 
left to right (across the horizon). These slices are so close 
together that we're going to make an assumption. Pick a slice in the 
middle, any one - it doesn't matter. Now look at it. It's going to 
look like a disk, right? Except that the bottom will be a little 
bigger than the top. What we do here is assume that because we made 
so many slices, the radius at the top and the bottom of this disk is 
the same. This is key! 

We can now use geometry to find the area of this disk because it is 
essentially just a cylinder with small height.  The volume of a 
cylinder is Pi * r^2 * h. What we have now is the volume of one tiny 
little disk. 
 
Now comes the fun part.  Do the above procedure for each remaining 
disk. See how long that could take? Luckily, there is a much easier 
way to add up the disks. It involves Riemann Sums and Integration, 
which you will study in Calculus. 

The procedure for the pyramid is similar: slice it up into small pieces. 
The difference? The pieces are not disks. They are now essentially in 
the shape of a TV dinner, a square base with small height. Find the 
volume (area of base * height) and add them up.  

I hope this clears up your question.

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