In this chapter we will use the word polygon to refer to a polygon together with its interior, even though properly we should use the term polygonal region. This should not cause any confusion.
Suppose one polygon is inside another. When treated as wire frames, the polygons would be considered as being disjoint; in the present context, they overlap. In general, if two figures share interior points, they will be considered as overlapping; otherwise, they will be considered as nonoverlapping.
We will associate with each simple polygon a nonnegative number called its area, and we will assume that area has certain reasonable properties.
Square brackets will be used to denote area. So, for example, the area of a quadrilateral ABCD will be denoted [ABCD].
Properties (ii) and (iii) certainly conform to our preconceived notions about area. We expect figures to have the same area if they have the same shape and size, and we also expect to be able to find the area of a large shape by summing the areas of the individual pieces making up the shape.
To develop ...