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Surface Area and Volume: Cubes and Prisms

Date: 05/27/98 at 15:11:19
From: Brittani
Subject: Math (surface area and volume)

What is the definition of surface area and volume?

How do you find the surface area and volume of a rectangular prism 
and cube?

Can you show me a picture of a rectangle prism and a cube?

What are the differences between surface area and volume? 
Are there similarities?

Date: 06/04/98 at 23:21:56
From: Doctor Peterson
Subject: Re: Math (surface area and volume)

Hi, Brittani. 

Let's start out with some general ideas about area and volume; then 
we'll look at prisms a bit.

A mathematician can get very picky about definitions, and sometimes 
the harder we think, the harder it gets to really define something 
like this. But I think what you probably want is just to understand 
what we mean when we talk about area and volume.

Basically, the surface area of an object means how much paper it would 
take to cover it (or how much paint, if you follow the directions and 
don't put it on too thick or too thin). The volume is how much clay it 
would take to make the object, or how much water it takes to fill it 
(if it were hollow). We measure area in "square somethings," such as 
square inches, which means if I cut a piece of paper into one-inch 
squares and try to paste them on the surface, how many would it take? 
Volume is measured in "cubic somethings," such as cubic inches, which 
means if I try to build the shape out of one-inch cubes, how many will 
it take to build it?

The main similarity between them is that both are measurements of the 
size of something. The main difference is that area deals only with 
the outside, while volume deals with the whole thing. Area is 
"two-dimensional" (like a sheet of paper, which doesn't have any 
significant thickness), and volume is "three-dimensional" (that is, it 
involves the height, width, and thickness). But when you're talking 
about surface area you have to be careful, because although the object 
you're measuring has three dimensions, you're just measuring its 
surface, which like a piece of paper is two-dimensional - it's just 
kind of bent a lot.

Here's a picture that can represent either a rectangular prism or a 
cube, depending on how your printer or screen print it:

      /          /|
    /          /  |
   +----------+   |
   |          |   |
   |          |   |
   |          |   +
   |          |   /
   |          | /

A cube is just a particular kind of rectangular prism, which is the 
same size in all three directions. A rectangular prism can be thought 
of as the shape you'd get if you put a rectangle flat on the table in 
front of you and then lift it straight up and imagine that it leaves a 
shape behind it as it goes. Or you could think of it as a stack of 
identical rectangles:

       /          / |
     /          / / |
   +----------+ / / |
   |----------| / / |
   |----------| / / |
   |----------| / / +
   |----------| / /
   |----------| /

To find the volume, you just multiply the three dimensions together. 
For example, if you have a 2 inch by 3 inch by 4 inch prism, the 
volume is 2 * 3 * 4 = 24 cubic inches. To see why, just imagine 
building it out of one-inch cubes. You'll need 6 (2 * 3) on the bottom 
layer, 6 on the next, and so on for four layers, so it will take 
6 * 4 = 24 cubes.

       /          / |
     /          /   |
   +----------+     | 4 inches
   |          |     |
   |          |     |
   |          |     +
   |          |   /
   |          | /  3 inches
    2 inches
But if you want the surface area, you have to figure out the area of 
each rectangular surface. There are a top and a bottom, both 2 * 3 
(6 square inches each), a front and a back, both 2 * 4 (8 each), and a 
left and right sides, both 3 * 4 (12 each), for a total of:
   12 + 16 + 24 = 52 square inches

Can you picture that? If not, get out some blocks and some paper and 
do it! Sometimes math can be fun.

If you like formulas, then for a prism that measures L units long by W 
units wide by H units tall, the volume is:

   L * W * H

and the surface area is:

   2 * L * W + 2 * L * H + 2 * W * H

Hope that helps.

For more about formulas and surface area of cubes and prisms, see the 
Dr. Math FAQ on Formulas for Geometric Figures:   

-Doctors Peterson and Sarah,  The Math Forum
Check out our web site!   
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