Surface Area of a Rectangular Solid

Date: 09/21/1999 at 19:34:32
From: Sarah Schuldenfrei
Subject: Surface area of solids

Hi,

My math teacher told us that we could calculate the surface area of a 
cube or a 3-dimensional rectangular object by calculating the area of 
the sides you can see and multiplying by 2. She said this didn't work 
for more complicated shapes. But I thought about it and I think it 
will work for any 3-dimensional object that you can build using cubes. 
Is this right?

Sarah

Date: 09/30/1999 at 13:25:20
From: Doctor TWE
Subject: Re: Surface area of solids

Hi Sarah. Thanks for writing to Dr. Math.

You have to be careful about how you're looking at the rectangular 
solid when applying your teacher's "shortcut." If you're looking 
directly at the center of one face, the shortcut won't work, because 
you only see that one face, like this:

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

If you're looking along the centerline of one face (but not at the 
center of the face), you'll only see two faces, like this:

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

Only if you're looking at the rectangular solid "off center" will you 
see 3 of the 6 faces, like this:

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

In this case, you teacher's shortcut will work.

The shortcut works due to symmetry. The three faces you don't see 
match one-to-one with the three faces you do see.

This won't necessarily work with other 3D figures, however, because 
they aren't necessarily symmetrical. Consider taking a cube made up of 
a block of 3x3x3 smaller unit cubes, and removing one edge cube from 
the hidden side, as diagrammed below. Each of the three faces you see 
has a surface area of 3x3 = 9 square units, for a total surface area 
of 3*9 = 27 square units. One of the hidden faces remains a 3x3 square 
(surface area = 9 square units), but the other two have been altered.

There are 8 unit squares along each of their flat faces, but where the 
other square was "removed," there are now four unit squares in the 
indented hollow spot. They are the three marked with an "x" in the 
diagram, plus one opposite the "9" on the lower left corner cube 
(marked 9,7,7). The total surface area of the figure is 
27+9+8+8+4 = 56, which is more than 2*27 = 54.


                  Front                          Back (hidden)

           +-----+-----+-----+                +-----+-----+-----+
          /  1  /  2  /  3  /|               /|  1  |  2  |  3  |
         +-----+-----+-----+3|              +3|     |     |     |
        /  4  /  5  /  6  /| +             /| +-----+-----+-----+
       +-----+-----+-----+2|/|            +2|/|  4  |  5  |  6  |
      /  7  /  8  /  9  /| +6|           /| +6|     |     |     |
     +-----+-----+-----+1|/| +          +1|/| +-----+-----+-----+
     |  1  |  2  |  3  | +5|/|          | +5|/|  7  | x  /|  8  |
     |     |     |     |/| +9|          |/| +9|     |---+x|     |
     +-----+-----+-----+4|/| +          +4|/| +-----+   | +-----+
     |  4  |  5  |  6  | +8|/           | +8|/  7  /  x |/  8  /
     |     |     |     |/| +            |/| +-----+-----+-----+
     +-----+-----+-----+7|/             +7|/  4  /  5  /  6  /
     |  7  |  8  |  9  | +              | +-----+-----+-----+
     |     |     |     |/               |/  1  /  2  /  3  /
     +-----+-----+-----+                +-----+-----+-----+

I hope this helps!

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