Remembering Area Formulas

Date: 12/23/2001 at 13:47:55
From: Summer
Subject: Areas

Hi, I was wondering if there was a good way to help me memorize 
the formulas for areas of different shapes. I am having a lot of 
trouble doing that and I need help. Is there a funny saying, or 
something?

Summer

Date: 12/23/2001 at 15:13:07
From: Doctor Ian
Subject: Re: Areas

Hi Summer,

You only need to memorize things that you don't understand, so feeling 
like you need to memorize something is a sign that you still don't 
understand it. 

And when you memorize things, it's easy to recall them incorrectly, by 
dropping a minus sign, or mixing up the numerator and denominator, or 
in a hundred other ways.  

For example, if you're trying to 'memorize' the associative, 
commutative, and distributive properties, you would be better off 
being able to reconstruct them from simple examples:

   Properties, Defined and Illustrated
   http://mathforum.org/dr.math/problems/rad.10.10.1.html   

Or if you're trying to 'memorize' the rules for working with 
exponents, you would be better off being able to reconstruct them from 
a few simple examples:

   Properties of Exponents
   http://mathforum.org/dr.math/problems/crystal2.01.22.01.html   

Understanding examples like these is kind of like keeping a spare key 
hidden under the ceramic frog next to your back door... you can use it 
to let yourself in if you ever lock yourself out of the house. 

If you'd like to write to me and tell me about some things that you're 
trying to memorize, I can try to help you understand them in a way 
that will make memorization unnecessary.  Would you like to try that? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/23/2001 at 15:57:24
From: Doctor Achilles
Subject: Re: Areas

Hi Summer,

Thanks for writing to Dr. Math.

I remember that I once learned the piano scales with "Every Good Boy 
Does Good" or something like that, but it never really stuck for 
me. I don't know of any funny saying or anything that can help you 
remember area formulas. But I think there is a better way to remember 
them than funny sayings.

First there are two things you have to memorize. Only two, though!

1) The area of a rectangle is width * length.

2) The area of a circle is pi * radius * radius.

Then, when you see any other area problem, just use those two facts to 
figure out the area. A square is easy enough to figure out. A square 
is just a special type of rectangle in which width = length.  So you 
just multiply side * side and you've got it.

For another example, you can find the area of a triangle:

h  |\
e  | \
i  |  \
g  |   \
h  |    \
t  |     \
   -------
    base

The area will be 1/2 * base * height.  How do I remember that?  Well, 
I imagine this rectangle:

   -------
h  |     |
e  |     |
i  |     |
g  |     |
h  |     |
t  |     |
   -------
    base

and I remember that the area will be base * height. Then I look at 
this triangle again with an imaginary rectangle drawn around it:

   ........
h  |\     .
e  | \    .
i  |  \   .
g  |   \  .
h  |    \ .
t  |     \.
   -------
    base

And the diagonal line cuts the rectangle in half. That means that the 
area of the triangle is half the area of the rectangle.

What about this triangle?

h       /\
e      /  \
i     /    \
g    /      \
h   /        \
t  /          \
   ------------
       base

Here I imagine another rectangle around it like this:

  ..............
h .     /\     .
e .    /  \    .
i .   /    \   .
g .  /      \  .
h . /        \ .
t ./          \.
   ------------
       base

Again, the area of the rectangle is base * height.

To convince yourself that the area of the triangle is half of that, 
draw the figure above on a piece of paper and cut out the rectangle.  
Then, cut the triangle out and save the two scrap pieces. If you put 
the two scrap pieces next to each other, they will add up to be the 
size of the triangle.

What about other strange shapes?

Here's a fun one, the parallelogram:

   --------------
h  \             \
e   \             \
i    \             \
g     \             \
h      \             \
t       \             \
         --------------
              base

To do this one, imagine a rectangle like this:

   --------------.......
h  \     .       \     .
e   \    .        \    .
i    \   .         \   .
g     \  .          \  .
h      \ .           \ .
t       \.            \.
         --------------
              base

The area of that rectangle is again base * height.  If you draw it and 
cut it out, you will see that the little area on the left that was 
left out of the imaginary rectangle is exactly the same size as the 
little area on the right that we added to the imaginary rectangle.  So 
the parallelogram's area is also base * height (Notice the difference 
here between base * height and side * side).

Here's one of the trickiest, the isosceles trapezoid:

          base1
         --------
h       /        \
e      /          \
i     /            \
g    /              \
h   /                \
t  /                  \
   --------------------
          base2

For this one, imagine a rectangle like this:

          base1
         --------.......
h       /.       \     .
e      / .        \    .
i     /  .         \   .
g    /   .          \  .
h   /    .           \ .
t  /     .            \.
   --------------------
          base2

Again, just like with the parallelogram, the area we left out on the 
left is equal in size to the area we added on the right. That means 
that the area of the trapezoid is the same size as the area of the 
rectangle we're imagining.

So far so good, but what is the area of the rectangle? Here's where 
trapezoids get a little bit tricky: the height of the imaginary 
rectangle is the same as the height of the trapezoid. But the base is 
bigger than base1 and smaller than base2. How big is it?

To help us figure out, let's look at the figure again:

          base1
         --------!!!!!!!
h       /.       \     .
e      / .        \    .
i     /  .         \   .
g    /   .          \  .
h   /    .           \ .
t  /     .            \.
  ---------------------
          base2

The !'s represent the part of the base that we added to base1. So the 
base of our imaginary rectangle is base1 + the length of the !'s.

So how long is that?  Look at this figure:

          base1
         --------!!!!!!!
h       /.       \     .
e      / .        \    .
i     /  .         \   .
g    /   .          \  .
h   /    .           \ .
t  /     .            \.
  #######--------------
          base2

The #'s represent the part of the base that we cut out of base2. So 
the base of our imaginary rectangle is base2 - the length of the #'s.

Here's the crucial point: The length that we cut off of base2 is 
EXACTLY EQUAL to the length we added to base1.  (Note that this is
true only for an isosceles trapezoid.  For other trapezoids, we end
up with the same formula, but we have to do a little more work to
get there.) 

So the base of the imaginary rectangle equals some number that is 
halfway between base1 and base2. How do you find a number halfway 
between? Just take the average. So the base of our imaginary rectangle 
is:

  base2 + base1
 ---------------
        2

So the area of our imaginary rectangle is:

            base2 + base1
  height * ---------------
                  2

And since the imaginary rectangle is the same size as the real 
trapezoid, that must be the area of the trapezoid.

Of course on a test you won't be able to go through ALL of that from 
scratch. So what you should do is practice making imaginary rectangles 
until you can do it quickly, then you should be able to go through 
without much trouble.

Plus, if you learn the reasons why areas equal what they do, instead 
of learning a funny rhyme, then if you ever forget the formulas 
completely, you can always figure them out again; but if you ever 
forgot the rhyme, you'd be stuck with nowhere to go. Learning math 
takes a lot of practice, but if you try to learn why things are the 
way they are, instead of just memorizing formulas with no reason 
behind them, you'll do better and you'll enjoy it more.

There are a lot more types of figures in geometry out there than I 
just told you about. Here's my last bit of advice: whenever you come 
to a new figure and you want to learn the area or volume or whatever, 
try to imagine a simpler figure drawn around it and try to understand 
how the size of the new figure is different from the simple one you 
imagined. That way you can understand the area of new figures without 
having to memorize a single thing you don't know already.  And if you 
ever come a figure you don't get, just write back and I'll try to help 
you understand it.

If there are other postulates and theorems that you're stuck on, try 
to understand why they are true and that should help you learn them.  
Feel free to write back if you have other questions about those as 
well.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/23/2001 at 20:15:26
From: Summer
Subject: Areas

I just wanted to say thank you! Your answers were both quite prompt 
and VERY indepth. I keep re-reading your advice and it makes so much 
sense to me! Thank you so much! I am going to be sure to tell all my 
friends who also need help in this area to be sure and contact you. 
I appreciate your taking the time to answer.

Summer