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Volume of a Sphere

Date: 05/19/2002 at 21:58:53
From: Maeve Mungovan
Subject: The volume of a sphere 

I know this topic has already been dealt with previously, and I began 
to understand it, but you have ommitted some vital information such 
as: Where does the 4 and 3 come from in (4/3)pi r^3? 

These issues have not been clearly treated. I have a math exam coming 
up in a week and it is extremely important that my question is 
answered. I do not wish to be short, but I am very stressed and your 
quick response would be much appreciated.

                   Thank you for your time and patience.

Date: 05/19/2002 at 23:18:35
From: Doctor Ian
Subject: Re: The volume of a sphere 

Hi Maeve,

I don't wish to be cryptic, but 'why' questions can be tricky.  
It's not always clear to the person answering the question 
exactly what counts as an answer to the person asking the 
question.  That's certainly the case here.  

One way to answer your question is:  If you slice a sphere up 
into very small pieces (using calculus), compute the volumes of 
the pieces, and add up those volumes, you come out with 

  v = (4/3) pi r^3

But it sounds as though that's not something you would consider a 
'proper' answer.  So let's try thinking of it another way.  

Imagine a sphere of radius r.  The smallest cube that it would 
fit around the sphere would have an edge length of 2r, right?  So 
the volume of that cube would be

  v = (2r)^3 = 8r^3

On the other hand, the largest cube that would fit inside the 
sphere would have a diagonal of 2r, which means that 

  sqrt(3) L = 2r

          L = 2r / sqrt(3)

where L is the edge length of the cube.  So the volume of that 
cube would be 

  v = (2r/sqrt(3))^3

    = ---------
      3 sqrt(3)

      8 sqrt(3)
    = --------- r^3

So if the volume of a sphere is 

  v = something * r^3

it must be the case that

   8 sqrt(3)
   --------- < something < 8

        1.54 < something < 8

and the actual value, 

   something = (4/3) pi = 4.19

is just about halfway in between.  Now, this doesn't say exactly 
why the coefficient of r^3 should be exactly (4/3)pi, but it does 
show that it's a plausible value.  But you might not accept this 
as an answer either. 

If neither of these explanations is what you're looking for, 
could you please write back and try asking the question in a way 
that makes it clear what you mean by 'why'? 

- Doctor Ian, The Math Forum 

Date: 05/25/2002 at 14:23:25
From: Maeve Mungovan
Subject: The volume of a sphere 

Actually, your response came just in time, and was greatly 
appreciated by both me and my younger sister. Thank you, your 
explanation was clear, precise and simple. Now I will be able to 
retain the formula for my upcoming examination. Thanks again,
Associated Topics:
High School Higher-Dimensional Geometry
Middle School Higher-Dimensional Geometry

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