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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

Megz
```

```
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

+ 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:

+ height * circumference

+ height * pi * diameter

+ height * pi * 2 * radius

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

+ height * 2 * radius)

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

+ 2 * height)

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

And now we can factor out a 2:

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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Polyhedra
Middle School Polyhedra

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