Connection Between Circumference and Surface Area

Date: 05/08/98 at 16:26:53
From: MARGOT DIAZ LEARNED
Subject: Derivation of the formula for surface area of a sphere

I am helping a 7th grader with his math and we got to this section in 
his book about the formula for the surface area of a sphere. It said 
you could not intuitively figure out the connection between the area 
of a circle and the surface area of a sphere and that you could look 
up the derivation of the formula in other texts. 

Anyway, we got interested in it, but I don't have any other geometry 
texts lying about the house, so I came here and found that someone had 
asked a similar question for which they received an answer from Dr. 
Anthony, but the explanation is beyond my son, and I was wondering if 
you could explain it for the more mathematically illiterate, such as a 
7th grader and a mom type?

Date: 05/19/98 at 11:21:12
From: Doctor Jeremiah
Subject: Re: Derivation of the formula for surface area of a sphere

Hi Margot:

You wanted to know about the connection between the area of a circle 
and the surface area of a sphere. Actually those two equations are not 
related. Instead the circumference of a circle has a connection to the 
surface area of a sphere and the area of a circle has a connection to 
the volume of a sphere.

Circumference is the space taken up by the line around the outside of 
the circle and surface area is the space taken up by the outside of a 
sphere. That is why they are related to each other.

The area of a circle is the space inside of the circle and volume of a 
sphere is the space taken up by the inside of a sphere.

The circumference of a circle is:     2*Pi*R
The surface area of a sphere is:      4*Pi*R^2
The area of a circle is:              Pi*R^2
The volume of a sphere is:            4/3*Pi*R^3

To understand the relationship between the equations you need to 
notice that all of them involve the radius R. The way you can tell how 
many R's to include (or what power to raise R to) is by how many 
dimensions the object you are measuring has.

When measuring the circumference of a circle, you are measuring the 
distance along a curved line, which is a one-dimensional curved line.  
If you look at the formula (2*Pi*R) there is only one factor of R in 
it, which corresponds with the one dimension.

When measuring the area of a circle, you are measuring something in 
two dimensions, which corresponds with the two R's in the 
formula: Pi*R^2.

It gets a little bit more complicated when we talk about surface area. 
While a sphere is definitely in 3 dimensions, its surface area is 
really only two-dimensional. Imagine measuring the surface area by 
wrapping the sphere in paper, and then cutting the paper off and 
flattening it out to measure how big it is. The surface area is
two-dimensional, just like this paper. Now look at the formula for 
the surface area: 4*Pi*R^2 - we're still in two dimensions.

So you can see that the difference is between related equations like 
circumference and surface area is that when you increase the number of 
dimensions from one to two or from you need to increase the exponent 
and find a new value for the coefficient at the front.

To understand why the number in front changes, think about when a 
square changes to a cube and the perimeter of the square changes to 
surface area of a cube:

   Perimeter of a square = 4L  
   (space around the outside of a square is 4 times the size of the 
   side which is a line)

   Surface area of a cube = 6*L^2  
   (space around the outside of a cube is 6 times the size of the side 
   which is a square)

See how the number in front changes? This is something that always 
happens. To understand what the value should be in any particular 
situation requires calculus.

Does that help? If you need more help please write again.

-Doctors Jeremiah and Sonya,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/