Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 5

AREA

5.1 Basic Properties

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.

5.1.1 Areas of Polygons

We will associate with each simple polygon a nonnegative number called its area, and we will assume that area has certain reasonable properties.

Postulates for polygonal areas:

(i) To each simple polygon is associated a nonnegative number called its area.

(ii) Invariance Property: Congruent polygons have equal area.

(iii) Additivity Property: The area of the union of a finite number of nonoverlapping polygons is the sum of the areas of the individual polygons.

(iv) Rectangular Area: The area of an a × b rectangle is ab.

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 ...