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### A Sphere's Surface Area and Volume

Date: 12/17/98 at 22:21:53
From: Matt
Subject: Relation between a sphere's surface area and its volume

What is the relation between a sphere's surface area and its volume?
I plotted a graph of (4 (pi)r^2)/(4/3 (pi)r^3), which is surface area/
volume. I need to know how this helps, if at all.

Date: 12/18/98 at 08:17:54
From: Doctor Jerry
Subject: Re: Relation between a sphere's surface area and its volume

Hi Matt,

Since S = 4*pi*r^2 and V = (4/3)*pi*r^3, the volume goes up as the cube
of the radius and the surface area as the square of the radius. Another
way of stating this is to calculate, as you have done, the ratio of the
volume to the area.

Some numerical illustrations would be useful:

Make a small table:

r       1        2         3        4          5

V       4.19     33.51     113.10   268.08     523.60

S       12.57    50.27     113.10   201.06     314.16

V/S     0.33     0.67      1        1.33       1.67

Do these values look familiar?

I'm going to introduce you to another relation, one you usually
wouldn't see until you get to calculus, so don't worry if you don't
quite understand what's going on. It will involve some large
calculations that would be best done by computer - it would take a long
time to do by hand. You might not notice this relation with small
values of r, but if you start with a large value of r, and then watch
how the volume changes (that is, compute V(r+1) - V(r)), you should get
some familiar numbers. Here is a table with r, S(r), and V(r+1) - V(r)
for r from 10,000 to 10,010:

r         S(r)             V(r+1) - V(r)

10000     1.25664 * 10^9     1.25676 * 10^9
10001     1.25689 * 10^9     1.25701 * 10^9
10002     1.25714 * 10^9     1.25727 * 10^9
10003     1.25739 * 10^9     1.25752 * 10^9
10004     1.25764 * 10^9     1.25777 * 10^9
10005     1.25789 * 10^9     1.25802 * 10^9
10006     1.25815 * 10^9     1.25827 * 10^9
10007     1.25840 * 10^9     1.25852 * 10^9
10008     1.25865 * 10^9     1.25877 * 10^9
10009     1.25890 * 10^9     1.25903 * 10^9
10010     1.25915 * 10^9     1.25928 * 10^9

For another way to look at this, compute (V(r+1) - V(r))/S(r) for large
values of r. You'll notice that this value gets closer to 1. This means
that as the radius increases, the volume increases by exactly the
surface area value.

As I said before, you're probably just looking for the first relation.
Consider the second one just as an introduction to some math you'll see
in the future.

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

Associated Topics:
High School Basic Algebra
High School Calculus
High School Geometry
High School Higher-Dimensional Geometry

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