Area and Volume

Date: 04/10/2002 at 18:54:55
From: Dex T.
Subject: Volume

Dear Dr. Math,

I am in seventh grade. I am currently in Pre-Algebra by Saxon Math.  
I can not figure out how to do volume. Please help me with a simple 
way to figure it out.

Thank you,
Dex T.

Date: 04/11/2002 at 14:40:01
From: Doctor Ian
Subject: Re: Volume

Hi Dex,

Do you understand length and area? Or are you a little sketchy on 
those, too? Length is pretty straightforward, but area can be a little 
tricky to understand at first. And volume is easiest to understand in 
comparison to area. 

A simple way to think about area and volume is this: Two things have
the same area if you would need the same amount of paint (or the same
amount of material) to cover them up. Two things have the same volume
if you would need the same amount of water to fill them up.

Note that two things can have different dimensions and still have the
same area. An easy way to demonstrate this is to take a sheet of
paper, like 

    +-----------+
    |           |
    |           |
    |           |
    |           |
    +-----------+

then tear it in half, and put the two halves together like 

    +-----------+-----------+
    |           |           |
    |           |           |
    +-----------+-----------+

Now, we have the same amount of paper, right?  So the area has to be
the same, even though the two shapes are different.  In fact, we could
do it again, 

    +-----------+-----------+-----------+-----------+
    |           |           |           |           |
    +-----------+-----------+-----------+-----------+

or we could rearrange the pieces to get something that isn't even a
rectangle, 

    +-----------+
    |           |
    +-----------+-----------+
    |           |           |
    +-----------+-----------+
                |           |
                +-----------+

If you could tear the paper into small enough pieces, you could even
rearrange them into something very close to a circle, and it would
still have the same area.  

Volume works the same way, except in three dimensions instead of two. 

If you go into a store like Lechter's or Williams-Sonoma and look at
measuring cups, you can find them in lots of different shapes: round,
square, short, tall, fat, thin.  But if two measuring cups are marked
with the same measurement - say, 1/3 cup - then they have the same
volume, because it takes the same amount of stuff to fill them.  That
is, you can fill one, and if you pour it into the other, it will fill
the other one too. 

So in a sense, area is a way to talk about 'covering', and volume is a
way to talk about 'holding'.  

Now, it turns out that, for lots of shapes, you can _compute_ the area
or volume, instead of having to measure it. That's a good thing,
because if you're trying to find the area of a building, you don't want
to have to paint it. Or if you're trying to find the volume of the
building, you don't want to have to fill it with water. 

Some things are just too big or too small to measure; or they may be
inconveniently located (e.g., in orbit around another planet, or at
the bottom of the sea); or they might be solid instead of hollow. 
Basically, there can be lots of reasons why you'd rather compute the 
area or volume of a thing rather than measuring it. 

Fortunately, many shapes - like circles, or triangles, or spheres, or
pyramids - have formulas that have been worked out. These formulas
let you plug in certain information about the shape, and get back the
area or the volume. And there are other techniques that can be used
to work out the volumes of more complicated shapes. 

Note that the formulas work in both directions. For example, if I know 
that each side of a square is 4 inches long, then I can use a formula 
to work out the area:

  area(square) = side * side

               = 4 inches * 4 inches

               = 16 square inches

However, suppose I know the area, and I want to know the length of a
side?  I can use the same formula to go in the other direction:

          area(square) = side * side

              16 sq in = side * side

 square root(16 sq in) = square root(side *side)

                  4 in = side

The same sort of thing works for volumes, too. For example, consider a 
rectangular prism, which is the shape of a cereal box. If you know the 
length, width, and depth, you can compute the volume of the box:

  volume(box) = length * width * depth

              = 8 inches * 4 inches * 2 inches

              = 64 cubic inches

But if you know the volume (perhaps because you measured it), and you
know two of the other dimensions, you can get the third:

                 64 cubic in = length * width * depth

                             = 8 in * 4 in * depth

                 64 cubic in = 32 square in * depth

  64 cubic in / 32 square in = depth

                             = 2 in

One of the tricky things about volume is that it's measured in lots of
different units. Usually when you _compute_ a volume, you end up with
'cubic units', like cubic inches, cubic centimeters, cubic feet, and
so on.  

However, usually when you _measure_ a volume, you use units like
ounces, cups, gallons, liters, and so on. So if you're going to work
with volumes a lot, you need to get used to converting back and forth
between these kinds of units.  But that's a subject for another
message. 

I hope this helps.  Write back if you'd like to talk more about this, 
or anything else.

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