Units and Square Units in Measuring Perimeter and Area

Date: 01/02/2005 at 14:46:37
From: Michelle
Subject: Unit vs. Square Units and Perimeter and Area

Why is perimeter measured in units whereas area is measured in squared 
units?

I understand that perimeter is one-dimensional whereas area is two-
dimensional, but I just don't understand why one is units while the  
other is squared units.

Date: 01/02/2005 at 17:37:20
From: Doctor Rick
Subject: Re: Unit vs. Square Units and Perimeter and Area

Hi, Michelle.

Let's be a bit clearer: Perimeter is measured in units OF LENGTH, 
while area is measured in units OF LENGTH squared.  For instance, 
perimeter can be measured in meters, while area can be measured in 
square meters.

Now imagine how you might measure a perimeter and an area physically. 
To measure the perimeter of a shape, you might take a string, lay it 
along the outer edge of the shape until you come back to the start of 
the string, and cut it off there.  Straighten out the string and 
measure its length: that's the perimeter of the shape.

Can you do the same with the area?  No!  You might try to fill the 
shape with string, but how much string it takes will depend on the 
thickness of the string--it's not just a matter of how long the string 
is.

Instead, we can imagine cutting a whole bunch of little squares--say,
one centimeter on each side--out of paper.  Then we can lay the 
squares side by side inside the shape until we have filled all the
space.  (We may have to cut up some of the squares into smaller pieces
and stick them in little corners.)  Count up all the squares we used,
and what do you have?  The area of the shape in SQUARE CENTIMETERS!

In other words, a shape with an area of, say, 25 square centimeters 
has an area 25 times as great as that of a square one centimeter on a 
side.  That's where the notion of "square units" comes from. 

Furthermore, the area of a rectangle (including a square) is the 
product of its length and width.  (This is easily confirmed by the 
method of cutting up little squares: they go in rows, and you can 
count them by multiplying the number of rows by the number in each 
row.)  So the area of the one-centimeter square is

  (1 cm) * (1 cm) = 1 cm*cm

The units are centimeters times centimeters.  What is a number times 
itself?  It's the square of the number.  So we call centimeters times 
centimeters "square centimeters".

How's that?  Does it answer your questions?  Let me know.

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