Finding the Perimeter and Area

Date: 12/01/98 at 21:00:55
From: Kottia
Subject: Perimeter and area?

Dear Dr. Math,

I'm a little confused over perimeter and area. I use addition to find 
the perimeter, and I use multiplication to find the area just the way 
my teacher taught me. I can do rectangles and squares, but not 
trapezoids, triangles and other funny looking shapes. We had to do this 
problem with a parallelogram which was 27yds at the top and bottom 
across, 13yds at the sides and 12yds in the parallelogram. I came up 
with 24yds for the area but the answer was really 324yds. Can you help?

Date: 12/02/98 at 12:06:29
From: Doctor Rick
Subject: Re: Perimeter and area?

Hi, Kottia. Perimeter isn't that hard if you remember that it is always 
the sum of the lengths of all the sides of the figure. Area is harder 
because you have a different formula for each kind of figure. You need 
to be careful to use the right formula for the figure, and to know the 
meaning of each quantity in the formula.

In your example of the parallelogram, the first mistake you made is to 
use the formula for a trapezoid. This formula is:

  Area of trapezoid = (top + bottom)/2 * height

You must be sure that top and bottom are the lengths of the parallel 
sides. You can actually use this formula for a parallelogram, because 
a parallelogram is a special kind of trapezoid with the top and bottom 
the same length. But you used length and width (bottom and side) 
instead of bottom and top.

The correct formula for a parallelogram is:

  Area of parallelogram = base * height

This formula works for rectangles and squares, too, because they are 
special kinds of parallelograms. But you must be careful not to confuse 
the length of a side with the height. These are the same for a 
rectangle or square, but not for most parallelograms.

Here is a diagram of your parallelogram:

   

What you should have done is this. The base is 27 yd. I assume that 
when you say "12 yd in the parallelogram" you mean the height - the 
length of a line joining the top and bottom and parallel to both of 
them. Then using the formula,

  Area of parallelogram = base * height
                        = 27 yd * 12 yd
                        = 324 yd^2

(Don't forget: area is measured in square units, like square yards 
[yd^2], not just yards!)

So once again, these are the main things to remember:

(1) Use the right formula for the figure - know the definitions of
    parallelogram, trapezoid, etc.!

(2) Know the definitions of the terms used in the formulas (base, 
    height, etc.) so you can use the right number for each. You can 
    find some of the important formulas here:

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

And one more: Don't be confused when a figure has more numbers than 
you need! You didn't need the length of the side to figure the area; 
that is only needed for the perimeter.

I hope this helps you.

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