Surface Area of a Cube

Date: 03/10/97 at 16:45:15
From: Aleah Boleman
Subject: Math (surface Area)

Dr. Mike,

Hi. I have a problem about surface area. I have to answer these 
questions for cubes with bases of 1, 2, and 3 units:
    
(a) Find the surface area of each cube.
(b) If the length of the cube is doubled, what is the surface area?
(c) If the length of the cube is tripled, what is the surface area?

Please help!

Date: 03/10/97 at 19:17:59
From: Doctor Mike
Subject: Re: Math (surface Area)

Dear Aleah,  
  
First think of what the surface of a cube looks like. The top is a 
square. The bottom is a square. The front and the other 3 sides are 
all squares.  

Let's take the cube with base 2 for example.  I think this means that 
if you measure along any edge, you find that it is 2 units long.  So 
each one of those squares on the surface is 2 units by 2 units.  How do 
you get the area?

          ______________________
          |                     |
          |                     |
          |                     | H
          |                     |
          |_____________________|
                    W  
  
To get the area of ANY rectangle, you multiply the dimensions, so
area = W*H in the example above.  Of course, W = H for a square, so 
the area of a 2-by-2 square surface is 2*2 = 4.  To get the total 
surface area of the cube, including top, bottom, front, back, left 
and right, you have that same square surface 6 times, so the total 
is 6*4 = 24 square units.  If units are inches, area is square inches. 
If units are miles then area is in square miles (a really big cube!). 

Now, for practice, you do the same thing for W = H = 1 and W = H = 3 
to get the surface area for the smaller cube and the larger cube that
have base edge lengths 1 and 3.
          
Okay, now for the rest of the problem.  You may have to help me a 
little on this one because I can see two different ways to interpret 
the problem.  When it is said that the "length of the cube is 
tripled", does that mean:
  
   1. That ALL the edge lengths are made 3 times as long so that
      the result is still a cube with squares all around?
OR
   2. That only ONE direction is stretched to make it 3 times as long,
      and it is not a cube anymore?  
  
You may have to ask your teacher for a clarification about this 
because I cannot tell for sure.  In order for you to be sure of 
getting full credit for your homework, you could do it BOTH ways so 
you will have it done no matter what the teacher says.
  
I will do one example so you know for sure what I'm saying.  Let's say 
we have the 2-by-2 cube.  Let's do it the way where we are just
stretching it in only one direction.  Let's say that the direction of 
stretching is up, so that it is 3 times as tall as it used to be. What 
do the bottom and top look like?  They are still squares, right? What 
do the 4 sides look like?  They used to be 2-by-2 but they were 
stretched to be 3 times as long, so they are now 2-by-6.  The 4 sides 
are not squares anymore, but they ARE rectangles, so we know how to 
find their areas, namely 2*6 = 12.  Now let's total up the areas of 
all 6 of the surfaces of the stretched cube: 
  
      Total area  =  4 + 4 + 12 + 12 + 12 + 12  =  56
  
If it turns out that "length tripled" really means that you just get
a bigger CUBE, then all of the six surface pieces will be squares of
the same size....but you know how to do that already.  
  
Now may be a good time for me to stop talking.  I think you know what 
the problem means, now, and you have seen some examples.  I think you 
are now qualified to finish it on your own.  Write back if something 
else leaves you puzzled.  

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/