Defining a Square Centimeter

Date: 04/09/2002 at 12:44:34
From: Young Kim
Subject: Geometry - Definition of 2-D Area

Dear Dr.Math,

I would greatly appreciate your help to answer my question.

I know that area is defined by the number of squares that cover a 
given section of the x-y plane. And we use unit square to cover the 
section to be measured. And books I've looked in say that the area of 
a unit square is 1 square centimeter because the two sides of the unit 
square are each 1 cm and we just multiply the two sides. But I cannot 
understand!

It seems to me that the whole point of measuring area comes down to 
measuring the area of a square of l cm length and width of 1 cm. And 
if we just multiply length and width to get the area of the square 
without knowing why, then we are back to the starting point!

My question is, Why is the area of a unit square the product of the 
two sides? By unit square, I mean a square of 1 cm length and 1 cm 
width.

Thank you for your help.
Kim

Date: 04/09/2002 at 12:58:59
From: Doctor Peterson
Subject: Re: Geometry - Definition of 2-D Area

Hi, Young Kim.

Before we can measure anything, we have to define the unit with which 
we will measure it. In this case, we define something called a square 
centimeter as the area of a one-centimeter square. That doesn't need 
proof, since the concept doesn't exist until we make this definition.

Given that definition, we can find the area of any rectangle (with 
integral sides, to start with) by laying out H rows of W unit squares. 
We can count them by multiplying, and find that the area is then W*H.

If we then apply this to the unit square, we of course get 1*1 = 1, 
but this just shows that our calculation is consistent with the 
definition. It also motivates our notation, where we write a square 
centimeter as cm^2 (with an exponent) because

    1 cm * 1 cm = (1*1)(cm*cm) = 1 cm^2

The fact that we are multiplying two centimeter measures makes it 
reasonable to call this area 1 cm^2. But again, that is not the 
definition of a square centimeter, only a conclusion from the 
definition.

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

Date: 04/13/2002 at 14:53:23
From: Young Kim
Subject: Geometry - Definition of 2-D Area

Dear Dr.Peterson,

Thank you so much for answering my question! Your crystal-clear 
explanation resolved my confusion once and for all. I really 
appreciate your help.

Best Regards,
Young Kim