Finding the Area of Squares and Triangles

Date: 06/20/99 at 22:06:00
From: Sky
Subject: I just don't get area

Hi -

Could you please explain area to me?

Sincerely,
Sky

Date: 06/21/99 at 18:19:16
From: Doctor Rick
Subject: Re: I just don't get area

Hi, Sky.

You have asked a broad question. I don't know just what about area is 
giving you trouble. Let's just go back to the basics - thinking about 
the basics may help you see harder problems in a new light.

We start thinking about area by considering squares. Suppose you have 
a tile floor whose tiles are 1 foot by 1 foot. (You might even want to 
find a tile floor, and at least pretend the tiles are 1 foot square.) 
We say that the area of each square is 1 square foot.

Mark out a rectangle on the floor with masking tape. Make this first 
rectangle right along the lines. Then you can find the area of the 
rectangle just by counting the tiles:

  +----+----+----+----+
  | 1  | 2  | 3  | 4  |
  |    |    |    |    |
  +----+----+----+----+
  | 5  | 6  | 7  | 8  |
  |    |    |    |    |
  +----+----+----+----+
  | 9  | 10 | 11 | 12 |
  |    |    |    |    | 5 feet
  +----+----+----+----+
  | 13 | 14 | 15 | 16 |
  |    |    |    |    |
  +----+----+----+----+
  | 17 | 18 | 19 | 20 |
  |    |    |    |    |
  +----+----+----+----+
        4 feet

This rectangle, 4 feet by 5 feet, has an area of 20 square feet - it 
contains 20 tiles. And of course you realize that this is 4 x 5. You 
can still find the area of a rectangle by multiplying the length by 
the width even if the sides aren't an integer number of feet:

  +----+----+----+----+--+
  | 1  | 2  | 3  | 4  |  |
  |    |    |    |    |  |
  +----+----+----+----+--+
  | 5  | 6  | 7  | 8  |  |
  |    |    |    |    |  |
  +----+----+----+----+--+
  | 9  | 10 | 11 | 12 |  |
  |    |    |    |    |  | 5.5 feet
  +----+----+----+----+--+
  | 13 | 14 | 15 | 16 |  |
  |    |    |    |    |  |
  +----+----+----+----+--+
  | 17 | 18 | 19 | 20 |  |
  |    |    |    |    |  |
  +----+----+----+----+--+
  |    |    |    |    |  |
  +----+----+----+----+--+
           4.5 feet

The area is 5.5 x 4.5 = 24.75 square feet. You can still count: 20 
whole tiles, 9 half tiles, and 1 quarter tile total 24 3/4 tiles.

The interesting part comes when we look at triangles. You would think 
diagonal lines would mess everything up, but they don't if you see the 
trick.

  +----------------------+
  |\                     |
  | \                    |
  |   \                  |
  |    \                 |
  |     \                |
  |       \              |
  |        \             |
  |          \           | 5.5 feet
  |           \          |
  |             \        |
  |              \       |
  |                \     |
  |                 \    |
  |                  \   |
  |                    \ |
  |                     \|
  +----------------------+
          4.5 feet

These 2 triangles are the same size and shape -- you could cut one out 
and place it exactly on top of the other. They have the same area. 
Both triangles together have an area of 5.5 x 4.5 square feet. One 
triangle alone has half this area, or (5.5 x 4.5)/2 = 12.375 square 
feet. The formula for the area of a right triangle is

     Area = (width x height) / 2

What if the triangle is not a right triangle? We can do just the same 
thing, with a little more cutting.

            B
  +---------+-------------------------+
  |        /|\                        |
  |  1    / |   \                     |
  |      /  |      \         2        |
  |     /   |         \               |
  |    /    |            \            |
  |   /  1  |               \         |
  |  /      |        2         \      |
  | /       |                     \   |
  |/        |                        \|
  +---------+-------------------------+
 A          D                          C

The area of triangle ABC is exactly half the area of the rectangle. 
How do I know? Make the cuts shown. The two triangles labeled "1" are 
the same size and shape, so they have the same area. The two triangles 
labeled "2" also have the same area. A "1" and a "2" make up the 
triangle. Two of each make up the rectangle. Therefore the rectangle 
has twice the area of the triangle. The formula is

     Area of ABC = 1/2 base x height

where the base is the length of the bottom of the triangle (the length 
of the rectangle), and the height is the length of BD (the height of 
the rectangle).

What is important to notice here is that BD is NOT on of the sides of 
the triangle. It is a line from the corner (vertex) that is not on the 
base, drawn down to the base, and PERPENDICULAR to it - just as the 
sides of the rectangle are perpendicular (at right angles) to the 
base.

We could go on to other shapes: parallelograms and trapezoids. If you 
forget the formulas for these, you can re-invent them for yourself by 
thinking jigsaw puzzles. Look at these "puzzles" and see if you can 
come up with the area on your own - you don't need the formulas.

  .......+---------------+------+
  :     /                |     /
  :    /                 |    /
  :   /           5 feet |   /   Hint: move the triangle on the right
  :  /                   |  /          to the left side as shown.
  : /                    | /
  :/                     |/
  +----------------------+
            6 feet

             4 feet
         +---------------+
        /|               |\                   /|\
       / |               |  \                / |  \
      /  |               |    \             /  |    \
     /   |5 feet         |      \          /   |      \
    /    |               |        \       /    |5 ft    \
   /     |   4 feet      |          \    /     |          \
  +------+---------------+-----------+  +------+-----------+
                   9  feet                       ?

Hint: Make the 2 side triangles into the one triangle as shown. What 
is the base of the big triangle? Now you have 2 areas to add together.

I haven't talked about circles, but everything else you have learned 
about can be solved by cutting in up into rectangles and triangles. 
This idea is much more important than memorizing formulas. You can't 
be sure you remembered the right formula for the shape, but you can be 
sure whether you're looking at a triangle or a rectangle.

I hope this helps you. If you have specific questions, please ask 
them - we're glad to help.

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