Surface Area and Volume of a Sphere

Date: 05/16/2000 at 22:31:39
From: Randy S.
Subject: Surface Area and Volume of a Sphere

Hello,

I am doing research for a project on why the coefficient in the 
formula for the surface area of a sphere is 4, and why 4/3 is the 
coefficient in the formula for the volume of a sphere. I have looked 
through your archives and have found that you can prove both by using 
an infinite number of pyramids inside a sphere, but is that the only 
geometric way? All the other methods used in the archives are too 
complicated for me (some use Calculus.)

I also have some trouble understanding Archimedes' hatbox because I 
don't understand why they multiply cosine and latitude lines. I also 
don't understand how one person proved it by using integrals because I 
don't even know what an integral is. So could you explain it to me 
geometrically, or else try to find some other geometric ways to 
explain why the coefficients are 4 and 4/3?

Date: 05/16/2000 at 23:15:55
From: Doctor Peterson
Subject: Re: Surface Area and Volume of a Sphere

Hi, Randy.

Don't worry about the integrals; that's calculus, and you'll need to 
learn a bit before you can follow the whole argument. The geometrical 
methods really use some of the ideas of calculus, but not the methods 
of calculus, so they can be followed more easily, though the work is 
harder.

It's not clear to me whether you saw this answer, which I think is the 
same as the hat-box, but doesn't mention cosines and latitudes, so it 
may be easier to follow:

   Volume of a Sphere
   http://mathforum.org/dr.math/problems/banijamal05.28.99.html   

There I start by finding the surface area using that method, and then 
use the pyramids to get the volume from that. 

There's another method that gets the volume directly using Cavalieri's 
theorem, which says that if every cross-section of two solids has the 
same area, then they have the same volume. We show that the sphere has 
the same volume as the cylinder circumscribed about the sphere, with a 
cone removed from each end.  See

   Volume of a Hemisphere Using Cavalieri's Theorem
   http://mathforum.org/dr.math/problems/pappas.9.09.99.html   

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