Area, Surface Area, and Volume: How to Tell One Formula from Another

Date: 01/18/99 at 22:37:23
From: Sandra Leslies
Subject: Surface area,area,volume

I am having trouble memorizing the geometric formulas. Say you have
the to calculate the volume of a pop can and you have the radius. How
can you tell that you are right and that you haven't done area?

Thanks.

Date: 01/19/99 at 13:20:43
From: Doctor Peterson
Subject: Re: Surface area,area,volume

Hi, Sandra. There's a nice simple answer to that last question, though 
it won't solve everything for you.

Suppose you vaguely remember that one formula for a cylinder is 
pi*r^2*h (pi times radius squared times height), and another is 
2*pi*r*h (twice pi times the radius times the height). You can tell 
which is the area and which is the volume by looking at the 
dimensions.

Suppose the radius is 2 inches and the height is 3 inches, and we
accept 3.14 for pi. Then our first formula gives

         pi*r^2*h = 3.14 * (2 in)^2 * (3 in)
                  = 3.14 * 4 in^2 * 3 in
                  = 37.68 in^3

Do you see how I work with the units just as if they were numbers (or 
variables in algebra), and end up with the units for the answer? Since 
the units in^3 are cubic, this is a volume.

Similarly, for the second formula

         2*pi*r*h = 2 * 3.14 * (2 in) * (3 in)
                  = 6.28 * 6 in^2
                  = 37.68 in^2

we get square units, in^2, so this is an area.

In general, you can just count the dimensions in the formula. r^2*h is 
the product of 3 dimensions, so it's a 3-dimensional quantity, a 
volume. r*h is the product of only two dimensions, so it's an area.

What this won't do for you is remind you about the constant terms --
is it pi or 2*pi? For a sphere, is it 4*pi*r^2 for area and 4/3*pi*r^3
for volume, or is it the other way around, 4/3*pi*r^3 for area and
4*pi*r^2 for volume? (The first pair is right!) When you get to
calculus, you'll learn a trick that helps. I'll show it to you now in
case it helps.

In calculus, there's a concept called the "derivative." In particular, 
the derivative of an expression like 

    a * x^n
    
is

    n * a * x^(n-1)       

Now, you're multiplying the a by what was the exponent n, and the new
exponenet of x is decreased by one. You don't need to have any idea
what a derivative is. It's easy to figure out, and that's all that
matters.

In simple cases, the derivative of a volume is the area -- 
specifically, the area of the surface where the volume is "growing" as 
you increase the variable for which you took the derivative. Here's an 
example: The volume of a sphere is

    4/3*pi*r^3

The derivative of this is

    3*4/3*pi*r^2 = 4*pi*r^2

which is the surface area of a sphere!

Here's a harder case: The volume of a cylinder is

    pi*r^2*h

If we take r as the variable that is "growing," the derivative is

    2*pi*r^1*h = 2*pi*r*h

which is the lateral surface area -- the side of the cylinder, where 
material would have to be added if you increased the radius.

If, instead, we take h as the "growing" variable, the derivative is

    pi*r^2*1*h^0 = pi*r^2

This is the area of one end of the cylinder, where you have to add 
material if the cylinder is growing in height. (Unfortunately, this 
doesn't give you the total area of both top and bottom.)

So this method, if you've been able to follow it, can help you check 
the numbers in a pair of formulas. You still have to memorize them, 
but at least you can check your memory.

Finally, probably the best way to learn these formulas is to know
where they come from. The sphere formulas, you may not be ready to
figure out on your own; but the cylinder formulas are simple. The
lateral surface area is just the circumference of the base circle
(2*pi*r) times the height (h) -- picture how you'd make the side of a
cylinder by rolling up a rectangle. The volume is the area of the base
circle (pi*r^2) times the height (h) -- just like the volume of a
rectangular solid.

So how can you memorize the formulas? I would suggest you write them 
all down in a table, and look for relationships. I've just told you 
how the formulas for a circle and a rectangle combine to give you a 
cylinder. The more of those you can find, the better. You'll also find 
some less obvious ones: the volume of a sphere and a cone have an 
interesting relationship. Make friends with the formulas, and they'll 
reveal some of their personal secrets to you.

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