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Helpful Ways to Remember Cylinder and Prism Formulas

Date: 03/28/2005 at 22:33:01
From: ed
Subject: Formulas get me confused

Hi -
  
I was wondering if you could tell me an easy way to memorize formulas.
I am stuck on memorizing formulas like volume, surface area, and 
lateral area for cylinders and prisms.

With all of the things I am learning, it is hard to remember 
everything.  I tried having my parents and friends help, but they 
didn't really help that much.



Date: 03/28/2005 at 22:53:56
From: Doctor Peterson
Subject: Re: Formulas get me confused

Hi, Ed.

I may be able to suggest ideas; but it will partly depend on how you
yourself learn best.  Think about something you learn easily; it might
be telephone numbers or sports figures or new words or friends' names.
How do you learn them?  Do you write them down, or recite them, or 
just use them a lot, or something else?  You'll probably want to find 
a way to use your best methods of learning for this purpose.

I myself learn something best when I understand it--where it came 
from, what it means, why it's true, how it is connected to other 
facts.  In fact, there are many formulas I don't really know as
formulas; I just know a picture or other fact from which I can build
the formula if I need it.

I tend to memorize just a few basic formulas, and then use those plus
an understanding of their meaning in order to extend them to other
cases.  You can see that approach here:

  Remembering Area Formulas
    http://mathforum.org/library/drmath/view/55418.html 

  Area, Surface Area, and Volume: How to Tell One Formula from Another
    http://mathforum.org/library/drmath/view/57791.html 

The surface area of any shape can be found by adding together the
surface areas of its parts.  The only one you can't figure out by 
doing that is the area of a sphere, so you just have to learn that 
one:

  S = 4 pi r^2

A cone is a little hard to figure out on your own, but it can be done;
it's explained here:

  Lateral Surface of a Cone
    http://mathforum.org/library/drmath/view/55082.html 

That one is fun to remember, though:

  S = pi r s

It's like the formula for the area of a circle, but with one of the
r's replaced with the slant height (which is like a radius, being the
distance from the apex to the circle).  Moreover, notice that pi r is
just half the circumference, so we can write the formula as

  S = 1/2 C s

And the formula for the lateral surface area of a regular pyramid is
actually the same: each triangular face has area equal to half its
base times its height, which is the slant height of the pyramid, and
putting it all together it comes to

  S = 1/2 P s

where P is the perimeter of the base.

Looking for similarities like this is a good way to memorize formulas!

I should point out that that one is the lateral surface area only--if 
you want the total surface area, you have to add on the area of the 
base.  I prefer to memorize only the formulas for individual parts 
like this, and add together the parts I need, which saves on memory.

For the other shapes--cube and cuboid (rectangular solid), prism, 
pyramid, cylinder--you can find the area just by breaking it into 
rectangles and triangles, so I don't bother memorizing the formulas at
all (unless I'm using them a lot and it helps to memorize them
temporarily).  The most interesting of these is the cylinder, which 
you might not think is so simple.  Picture a soup can.  Cut off the 
top and bottom--those areas are just circles, so you know the formula 
for them already.  Now cut down the seam on the side, and flatten the
lateral surface out.  What does it look like?  A rectangle!  Its width 
is the circumference of the circle, 2 pi r, and its height is the 
height of the can.  So you can work that out:

  S = 2 pi r h

The prism is similar.  The top and bottom are polygons (such as
triangles or rectangles), so you can do them; and the sides are made
up of rectangles.  In fact, if you treat the lateral surface of a 
prism like my cylinder and cut it apart only along one seam, you'll 
find that the whole thing is just one rectangle again, whose width is 
just the perimeter of the base polygon! So the formula is really the 
same as for the cylinder:

  S = P h

where P is the perimeter.  This is just the lateral surface, of 
course.

Now, volumes are really easier.  Apart from the sphere, which again is
special, there are just two cases to learn.  First, the prism and
cylinder have this formula:

  V = B h

where B is the AREA of the base and h is the height.  Then, the 
pyramid and the cone have this formula:

  V = 1/3 B h

So if you just remember that when there's a point at the top the 
volume is 1/3 of what it would be with a flat top, you've got the formula.

(Interestingly, the area of a triangle is similarly 1/2 of a rectangle
with the same base and height.  So for two dimensions, a point makes 
the AREA half as much, and in three dimensions, a point makes the
VOLUME one third as much.  That helps to remember the 1/3 part.)

The second link I gave above mentions one fact that helps a lot in
keeping track of these things: any area formula involves multiplying
two lengths, while any volume formula has you multiply three lengths
(or an area times a length).

Here is another page of interest, discussing the concepts of volume
and surface area for simple prisms:

  Surface Area and Volume: Cubes and Prisms
    http://mathforum.org/library/drmath/view/55016.html 

I know I've said a lot, and it may seem like too much, but take it
slowly and work your way through your list of formulas with these
ideas in mind.  Find your own patterns and learn how to put the basic
formulas together, and it will start to feel simpler!

Let's make a quick list:

                    Lateral area      Volume
                    ------------   ------------
Prism or cylinder          P h              B h
  Cylinder            2 pi r h         pi r^2 h

Pyramid or cone        1/2 P s          1/3 B h
  Cone                  pi r s     1/3 pi r^2 h

Sphere                4 pi r^2     4/3 pi r^3

where

  h = height            P = perimeter of base
  s = slant height      B = area of base
  r = radius of base

You don't really have to learn the special formulas for the cylinder
and cone, but they're nice enough to be worth knowing.

Let me know if you need any more help.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Higher-Dimensional Geometry
Middle School Higher-Dimensional Geometry

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