Surface Area Of Three Cubes Glued Together

Date: 10/22/2002 at 17:50:49
From: Ben Johns
Subject: Surface Area On Cubes

You have 3 cubes, one with an edge measuring 2, one 6, and one 8.  
If you glue them together so you have the smallest possible surface 
area, what will that surface area be?

I tried putting blocks together to figure it out and I came up with 
536, but I'm not completely sure. We have to come up with other ways 
to solve the problem to confirm our answer. I can't think of other 
ways to solve it. I already put the blocks together and used surface 
area formulas, but I don't know if I'm right because I have no clue 
what to do for a confirmation.

Thanks for your help.

Date: 10/22/2002 at 18:31:37
From: Doctor Ian
Subject: Re: Surface Area On Cubes

Hi Ben,

The first thing you need to do is make sure you've considered _all_ 
the possibilities. 

With two cubes, as long as the smaller face is completely in contact 
with the larger one, that's the best you can do. The visible surface 
area (SA) is then

    SA of larger cube 
  + SA of smaller cube 
  - (2 * area of one face of smaller cube)

Does that make sense?   

What about that third cube?  You could put it on a side of one of the 
other cubes, and that would hide twice the area of one of its sides.  
Can you do better?

If the idea is to hide faces, then the best you could do would be to 
make sure that the smaller cube shares a face with _each_ of the 
other cubes. Now the visible surface area is 

    SA of large cube 
  + SA of middle cube 
  + SA of small cube
  - (2 * area of one face of middle cube)
  - (4 * area of one face of small cube)

For a cube with edge length L, the surface area is 6L^2. So this works 
out to 

    6(8^2)
  + 6(6^2)
  + 6(2^2)
  - (2 * 6^2)
  - (4 * 2^2)

Which, happily, adds up to 536 square units. 

Does this help? 

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