Derivation of Sphere Volume and Surface Area Formulas

Date: 03/14/2005 at 16:10:28
From: Alex
Subject: Why the formula for the volume of a Sphere has '4/3' in it

Currently in math class we are discussing surface areas and volumes of
solids.  I would like to know why the volume formula for the sphere is
(4/3)*pi*r^3 and why the surface area formula is 4*pi*r^2.

I understand why the volume formula contains pi, and radius, and is 
cubed, but I do not understand why 4/3 must be in the formula.  The 
same goes for the surface area formula.  I can understand the pi*r^2, 
but I do not understand why it is 4 and not another number.

I'm sure that both the (4/3) and the 4 were derived some way, but I 
cannot understand why it is so.  I often question formulas like this.

Date: 03/15/2005 at 08:23:21
From: Doctor Rick
Subject: Re: Why the formula for the volume of a Sphere has '4/3' in it

Hi, Alex.

I appreciate your desire to understand the reasons for things and not 
just take a teacher's word for it.  That is an important trait in a 
mathematician.  Unfortunately, it will be a while before you have 
enough math under your belt to understand the proof in full detail.  
It involves calculus.

For now, we can help you see that it makes sense, though, even if we
can't show that it *must* be true.

Let's start with the surface area.  Take a sphere of radius R, and 
imagine constructing a cylindrical box with radius R and height 2R.  
You can see that the sphere will fit snugly inside this box.

Archimedes, the Greek mathematician, proved a surprising fact: the 
surface area of the sphere is exactly the same as the lateral surface 
area of the cylinder (that is, the surface area not including the two 
circular ends).  You can calculate the lateral surface area of the 
cylinder and you will see that it is 4*pi*R^2.  The following item in 
the Dr. Math Archives describes what Archimedes did to prove this 
result:

  Volume of a Sphere
    http://mathforum.org/library/drmath/view/55135.html 

Also on that page you will see an explanation of the 4/3 in the volume 
of the sphere.  In brief, you can imagine drawing a tiny triangle on 
the surface of the sphere and connecting its corners to the center of 
the sphere.  You have made a very narrow pyramid.  The volume of a 
pyramid is 1/3 times the area of the base times the height.  Thus the 
volume of this pyramid is 1/3 times the radius of the sphere, times 
the area of that little triangle.

Now, imagine that you cover the sphere with tiny triangles, and thus 
cut the sphere into millions of narrow pyramids.  The total volume of 
the pyramids is 1/3 times the radius of the sphere times the sum of 
the areas of the tiny triangles.  In other words, the volume of the 
sphere is 1/3 times R times the surface area of the sphere!

  V = (1/3)R * 4*pi*R^2
    = (4/3)pi*R^3

How's that?

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

Date: 03/15/2005 at 19:39:00
From: Alex
Subject: Why the formula for the volume of a Sphere has '4/3' in it

Thank you for your answer.  Unfortunately, I believe that there is an 
error.  When I looked up the formula for the lateral area of a 
cylinder, I found that it is 2*pi*r*h or pi*d*h.  The formula that you 
used to explain why it is (4/3)*pi*r^3 contains the surface area of a 
sphere, not a cylinder, in it.  The formula makes sense, and so does 
the explanation why, but it is confusing that the surface area of the 
sphere is interchanged with that of the cylinder.  As far as I 
understand, this is impossible.  I thank you in advance for your time.

Date: 03/16/2005 at 08:15:18
From: Doctor Rick
Subject: Re: Why the formula for the volume of a Sphere has '4/3' in it

Hi, Alex.

Didn't I say this fact was "surprising"?  I'm glad you agree!  To me, 
it's pretty amazing.  In fact, Archimedes thought it was so wonderful 
that he had a carving of a sphere and a cylinder put on his tombstone.

The surface area of a sphere is 4*pi*r^2.  The lateral area of a 
cylinder is the height times the circumference of the base, or
2*pi*r*h.  For a cylinder of height 2r, that makes

  S = 2*pi*r*(2r)
    = 4*pi*r^2

Whether you think it's impossible or not, it's true!  The surface area 
of a sphere is the same as the lateral surface area of the cylinder 
into which the sphere fits.

I pointed you to a page where we describe Archimedes' method of 
proving this:

  Volume of a Sphere
    http://mathforum.org/library/drmath/view/55135.html 

That page, like my explanation, begins with the surface area, then 
uses that to show that the formula for volume of a sphere makes sense.  
Doctor Peterson (who happens to be my twin brother) talks about 
slicing through both the sphere and the cylinder to make lots of very 
thin slices.  Then he demonstrates that the surface area of each slice 
of the sphere is equal to the surface area of the corresponding slice 
of the cylinder.  Though the radius of the sphere slice is less than 
the radius of the cylinder slice, the surface of the sphere slice is 
sloped, which makes it longer (measuring along the slope) than the 
surface of the cylinder slice.  These two effects happen to cancel
out: the factor by which the radius is reduced is the reciprocal of 
the factor by which the length is increased, so the surface areas of 
the two slices come out the same. And if the areas of the slices are 
the same, then the areas of the whole sphere and the whole cylinder 
(without the ends) are the same.

You'll have to go through Doctor Peterson's explanation to see that 
this is true.  I can't make it any simpler than that.  I'll be glad to 
answer more questions, though, until you're satisfied.  Just remember 
that *full* satisfaction won't come until you get to calculus.

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

Date: 03/16/2005 at 14:58:54
From: Alex
Subject: Thank you (Why the formula for the volume of a Sphere has
'4/3' in it)

Thank your very much for your help.  I now understand where the 
surface area of a cylinder and sphere can be the same.  I was confused 
because I thought that you were saying that the surface area of a 
cylinder and sphere is always exactly the same.  Now I understand.  
Thank you very much!

Sincerely,

Alex