Net of a Box

Date: 05/23/99 at 14:31:17
From: Holley Goodwin
Subject: Drawing a flat pattern of a box

Hi, 

I am a 7th grader, and my teacher asked me to do a project to present 
to the class. She asked me to come up with some dimensions (length, 
width, and height) and find the surface area and volume. Then I was 
told to draw a flat pattern of the box, I don't know how to do any of 
that. Can you please help me?

Date: 05/24/99 at 13:11:38
From: Doctor Peterson
Subject: Re: Drawing a flat pattern of a box

Hi, Holley.

Here's an idea for making this an impressive (and helpful) project.

Find some kind of small cubes - maybe baby blocks left over around 
your house, or dice, or something like that. Measure them and see how 
many you have. Then you can actually show the volume of the box you 
make by putting that many blocks inside it; and you can show the area 
by pasting the right number of paper squares on the outside of it.

Let's suppose you want to make a box like this, with dimensions in 
"blocks" (that is, using the edge of your blocks as a unit of length; 
I'll suppose that the blocks are one-inch cubes):

              EMPTY BOX

        +-------------------+
       /                   /|
      /                   / |
     /                   /  |
    +-------------------+   | 4
    |                   |   |
    |                   |   |
    |                   |   |
    |                   |   +
    |                   |  /
    |                   | /
    |                   |/ 2
    +-------------------+
               5

To find the volume, you will have to count the number of blocks you 
can fit. If you make one layer on the bottom, it will take 5x2 = 10 
blocks:

     BOX WITH ONE LAYER OF BLOCKS

        +-------------------+
       /                   /|
      /                   / |
     /                   /  |
    +-------------------+   | 4
    |                   |   |
    |   +---+---+---+---+---+
    |  /   /   /   /   /|  /|
    | +---+---+---+---+-|-+ +
    |/   /   /   /   /  |/|/
    +---+---+---+---+---+ +
    |   |   |   |   |   |/ 2
    +---+---+---+---+---+
              5

Four layers like this will take 4x16 = 64 blocks: 

    BOX WITH FOUR LAYERS OF BLOCKS

        +---+---+---+---+---+
       /   /   /   /   /   /|
      +---+---+---+---+---+ +
     /   /   /   /   /   /|/|
    +---+---+---+---+---+ + + 4
    |   |   |   |   |   |/|/|
    +---+---+---+---+---+ + +
    |   |   |   |   |   |/|/|
    +---+---+---+---+---+ + +
    |   |   |   |   |   |/|/
    +---+---+---+---+---+ +
    |   |   |   |   |   |/ 2
    +---+---+---+---+---+
              5

Now to make a flat pattern, you need to lay out the six sides of the 
box so they will fold up. Here's one way:

                       5
             +-------------------+
             |                   |
             |        top        |2
         2   |                   |   2
     +-------+-------------------+-------+
     |       |                   |       |
     |       |                   |       |
     |       |                   |       |
     | left  |       front       | right |4
     |       |                   |       |
     |       |                   |       |
     |       |                   |       |
     +-------+-------------------+-------+
             |                   |
             |      bottom       |2
             |                   |
             +-------------------+
             |                   |
             |                   |
             |                   |
             |       back        |4
             |                   |
             |                   |
             |                   |
             +-------------------+

Now you can find the surface area by adding up the areas of the six 
sides, each of which you can find by multiplying its length times its 
width.

Even if you don't actually fill it with cubes (which would require 
very careful measurements), just thinking of it this way should help 
you understand what it's all about.

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