Tuesday, October 5, 2010

Basic Idea of Polygon

In this chapter, we will explain about polygon. We may start from the basic ideas of a polygon, the convexity, basic theorems of polygon, regular polygon, some kinds of polygon, until kinds of quadrilateral. We give you two kinds of exercise in this chapter, because of its large subject material. Some our daily mistakes are also given and directly solved here. We wish for next activity, the mistakes will not be happen again.


Some figure which is polygon is shown below.

Figure 4.1 Polygon

Some figures which are not polygon are shown below.

Figure 4.2 Not polygon

We shall give the definition of a polygon in a moment. In these figures, observe that a polygon consists of line segments which enclose a single region.

A four-sided polygon is called a quadrilateral (Figure 4.1(a) or (f)). A five-sided polygon is called a pentagon (Figure 4.1(b)), and a six-sided polygon is called a hexagon (Figure 4.1(e)). If we kept using special prefixes such as quad-, penta-, hexa-, and so on for naming polygons, we would have a hard time talking about figures with many sides without getting very confused. Instead, we call a polygon which has n sides an n-gon.

For example, a pentagon could also be called a 5-gon; a hexagon would be called a 6-gon. If we don't want to specify the number of sides, we simply use the word polygon (poly- means many). As we mentioned for triangles (3-gons), there is no good word to use for the region inside a polygon, except "polygonal region", which is a mouthful. So we shall speak of the area of a polygon when we mean the area of the polygonal region, as we did for triangles.

If a segment PQ is the side of a polygon, then we call point P or point Q a vertex of the polygon. With multisided polygons, we often label the vertices (plural of vertex) P1, P2, P3, etc. for a number of reasons. First, we would run out of letters if the polygon had more than 26 sides. Second, this notation reminds us of the number of sides of the polygon; in the illustration, we see immediately that the figure has 5 sides:

Figure 4.3 A Hexagon

Finally, if we want to talk about the general case, the n-gon, we can label its vertices P1, P2 , P3 , … ,Pn - 1, Pn as shown:

Figure 4.4 General n-gon

We can now define a polygon (or an n-gon) to be an n-sided figure consisting of n segments

which intersect only at their endpoints and enclose a single region.

Experiment 4.1.
Below are two rows of polygons. Each polygon in the top row exhibits a common property, while those in the bottom row do not.

Figure 4.5 Polygons

Can you discover what the top row polygons have in common that the bottom ones do not? Try to state the definition of your property as specifically as possible, using terms and concepts that we have already defined. There are many possible answers.

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