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Finding Surface Areas

Date: 06/17/2002 at 18:35:28
From: Megz
Subject: Measurement.

I don't understand the formulas to figure out surface area on the 
different kinds of figures. It is really confusing to me, and hard 
to explain about what I don't understand. It's basically the whole 
thing about it.


Date: 06/17/2002 at 21:19:27
From: Doctor Ian
Subject: Re: Measurement.

Hi Megz,

The basic idea is that to find the surface area of a figure, you 
break the figure up into individual sides or 'faces'; find the 
area of each face; and add them all up.

Sometimes this results in a very compact formula that doesn't 
look very much like you did that; but it's still what's going on.

Let's look at a couple of examples.  How about a cube?  There are 
six faces to a cube.  (If you forget this, recall that the sides 
of dice are numbered 1 to 6.)  

Each face of a cube is a square, and the length of each side of 
the square is the same as the length of an edge of the cube.  The 
area of a square is the length of a side multiplied by itself.  
So the surface area of a cube is 

  surface area =    area of face 1
                  + area of face 2
                  + area of face 3
                  + area of face 4
                  + area of face 5
                  + area of face 6

               =    edge*edge
                  + edge*edge
                  + edge*edge
                  + edge*edge
                  + edge*edge
                  + edge*edge

               = 6 * edge * edge

               = 6 * edge^2

Now, suppose we have, not a cube, but a rectangular prism (like 
the shape of a cereal box).  We still have six sides, but now 
they come in pairs, and each side is a rectangle.  Two of the 
rectangles have dimension width * height; two have dimension 
width * length; and the remaining two have dimension length * 
height.  So the surface area is 

  surface area =    area of face 1
                  + area of face 2
                  + area of face 3
                  + area of face 4
                  + area of face 5
                  + area of face 6

               =    width*height
                  + width*height
                  + width*length
                  + width*length
                  + length*height
                  + length*height

               = 2 * (width*height + width+length + length*height)

Let's look at one more example:  A cylinder.  There are three 
'faces' to a cylinder: a circular one at each end, and the big 
curved side.  

The area of each circle is pi times the square of the radius.  So 
the surface area is 

  surface area =   area of circle 1
                 + area of circle 2
                 + area of side

               =   pi * radius^2
                 + pi * radius^2
                 + area of side

What about the side?  Well, imagine that you make a cut from top 
to bottom, and unroll the side.  You get a rectangle, right?  The 
height of the rectangle is the height of the whole cylinder.  
What is the width of the rectangle?  It's the circumference of 
the circles!  So we can complete the formula:

               =   pi * radius^2
                 + pi * radius^2
                 + height * circumference

               =   pi * radius^2
                 + pi * radius^2
                 + height * pi * diameter

               =   pi * radius^2
                 + pi * radius^2
                 + height * pi * 2 * radius

Now, each of these terms has pi in it, so we can factor that out:

               =  pi * (  radius^2
                        + radius^2
                        + height * 2 * radius)

Each term in parentheses also has a radius in it, so we can 
factor that out too:

               =  pi * radius * (  radius
                                 + radius
                                 + 2 * height)

We can add the radii together:

               =  pi * radius * (2 * radius + 2 * height)

And now we can factor out a 2:

               = 2 * pi * radius * (radius + height)

Now, here's the thing.  If you're not going to use this formula 
every day, there's absolutely no point in memorizing it.  I 
certainly haven't!  If I want to compute the surface area of a 
cylinder, I'll break it into two circles and a side, compute 
those areas, and add them up.  And I recommend that you do the 
same thing, rather than trying to learn the compact formulas.  

Does this help? 

- Doctor Ian, The Math Forum
Associated Topics:
High School Polyhedra
Middle School Polyhedra

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