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### Surface Area of a Cube

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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?

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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?

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/
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Associated Topics:
Middle School Geometry
Middle School Polyhedra

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