Area and Perimeter

Date: 05/01/2001 at 21:37:12
From: Jessica L.
Subject: Area and Perimeter

I do not understand area and perimeter.

Date: 05/02/2001 at 13:44:09
From: Doctor Ian
Subject: Re: Area and Perimeter

Hi Jessica,

The word 'perimeter' literally means 'distance around'. Think about a 
rectangle like this one:

        3 ft
     A---------B
     |         | 2 ft
     |         |
     D---------C

One way to 'walk around' the rectangle would be to move from A to B 
(a distance of 3 ft.), then from B to C (a distance of 2 ft.), then 
from C to D (a distance of 3 ft.), and finally from D to A (a distance 
of 2 ft.).

The total distance involved would be 3 ft + 2 ft + 3 ft + 2 ft, or 
10 ft. So that's the perimeter of the rectangle: 10 ft.

Area is more complicated, because it involves two dimensions, whereas 
perimeter involves only one. The way I always think of area is in 
terms of the amount of paint that I would need to cover a shape. If 
something has twice as much area, then I'd need twice as much paint.

For a rectangle, we compute area by multiplying the length by the  
width:

        3 ft
     A---------B
     |         | 2 ft       perimeter = 3 + 2 + 3 + 2 = 10 ft
     |         |                 area = 3 * 2 = 6 square feet
     D---------C


If we double the length of each side, we get twice the perimeter, but 
_more_ than twice the area:

             6 ft
     A-------------------B
     |                   |
     |                   |         perimeter = 6 + 4 + 6 + 4 =20 ft
     |                   | 4 ft
     |                   |              area = 6 * 4 = 24 square feet
     |                   |
     D-------------------C


A lot of people get confused about that point, but a diagram can help 
make things clearer:

                    _a_
                   /   \

            +-----+-----+
         /  |     |     |  \ b
        |   |     |     |  /
     2b |   +-----+-----+ 
        |   |     |     |
         \  |     |     |
            +-----+-----+

            \__________/
                 2a

For a rectangle, if I double the length of each side, I get four times 
the area (but twice the perimeter).

Note that we can have more than one rectangle with the same perimeter, 
but different areas:

          3                   4
     A---------B       A-------------B     
     |         | 2     |             | 1   
     |         |       D-------------C     
     D---------C

       p = 10               p = 10
       a = 6                a = 4

In fact, for a given perimeter, we can make the area as close to zero 
as we'd like, by making the rectangle long and thin:

                 4.9999999
     A--------------...-------------B
     |                              | 0.0000001
     D--------------...-------------C

                 p = 10
                 a = .0000005

For polygons (triangles, pentagons, hexagons, and other shapes that 
you make by linking line segments together) perimeter is always pretty 
easy to compute (you just add up the lengths of the sides), but 
computing area gets more complicated. For a triangle, the formula is:

     area = (1/2) * length_of_base * height

For a trapezoid, the formula is:

            top_length + bottom_length
     area = -------------------------- * height
                       2

Note that for rectangles, finding the height is trivial - you just 
choose any side to be the height. For other shapes, it can be a little 
more involved, and often requires the use of the Pythagorean theorem 
(which is one of the reasons that your teachers want you to learn it).

The Dr. Math FAQ has a 'formulas' section that contains formulas for 
area for most of the shapes that you'll come across in your math 
classes:

  http://mathforum.org/dr.math/faq/formulas/   

Does this help? Write back if you'd like to talk about this some more, 
or if you have any other questions.

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