Associated Topics || Dr. Math Home || Search Dr. Math

### 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

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.

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
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search