Area 1

Note: The following information are referenced to my previous knowledge and are not entirely my independent research.

We think we are quite familiar with the concept of area, or the 2 dimensional "size" of a shape cover over a plane (say it is the surface of the earth, or the table, or paper, or R X R plane).

But what is area? How do we REALLY measure area? Why are formulas for areas so, weird looking? Some of the questions will be solved below, but not all. To do this we have to DEFINE what the heck IS area.

First, we have to define shapes. We say by shape we mean the set of ALL, "closed", 2 dimensional graphs. I cannot establish a clear concept of "closed", we can consider closed as meaning there is a finite region within the graph.

Now, we define a square, a square is a shape with 4 identical sides, and the angle between any two adjacent sides is pi/2 radian. Why we use radian? We will see in later sections. Further, we call a square with side 1 a unit square.

Now we DEFINE the area of a UNIT SQUARE as 1 square unit. You can think of it as its side times side, but that's not it----- the area is defined for it.

Then we say the area of a shape is the total number of UNIT SQUARES that fits in the closed region (as the concept of closed is not perfect, you can assume we are dealing with common shapes).

We will work step by step on some example to show this definition works really well in the next section.