Chapter 8

Introductory Geometry

Section 8.2

Polygons

Closed Curves and Simple Closed Curves

Closed curves are figures that can be drawn so that you begin and end at the same place. The “curve” can be either a straight line or a curve. A simple closed curve is a closed curve that does not cross over itself.

Closed curve, not simple

Simple curve, not closed

Curve, not simple and not closed

Simple closed curve

Polygons

Polygons are simple closed curves whose sides are all line segments. The points that join the line segments are called the vertices of the polygon.

polygons not polygons

Polygons are named by the number of sides that make them up. The prefix before the suffix “agon” indicates the number of sides.

Sides Prefix Name Shape

3 Tri Triangle

4 Quad

Tetr

Quadrilateral

Tetragon

5 Pent Pentagon

6 Hex Hexagon

7 Hept Heptagon

8 Oct Octagon

9 Non Nonagon

10 Deca Decagon

Geoboards

A geoboard is a square board with pegs laid out in a grid pattern. In our case we show them as an array of dots in a grid pattern. You can connect the dots with straight lines to make different types of geometrical objects.

pentagon octagon quadrilateral

Diagonals

A diagonal of a polygon is a line segment that connects two vertices of the polygon that is not part of the polygon itself already. The diagonals can be either inside or outside the polygon.

triangles have 0 diagonals quadrilaterals have 2 diagonals pentagons have 5 diagonals

hexagons have 9 diagonals

n n-3 n(n-3) n(n-3)/2 sides diagonals

3 0 0 0 3 0

4 1 4 2 4 2

5 2 10 5 5 5

6 3 18 9 6 9

We use the table below to reason inductively to get the formula for the number of diagonals in a polygon.

A polygon with n sides has:

2

)3( nn diagonals

Convex Polygons

A simple closed curve divides the plane it is in into 3 parts; the inside, the curve itself and the outside (this is known as the Jordan Curve Theorem). Since a polygon is a type of simple closed curve this idea applies. A polygon is called convex if all of its diagonals are inside the polygon. If there is even one diagonal that is outside the polygon it is not convex.

Convex Polygons Not Convex Polygons