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
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
Shape
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.
quadrilaterals have 2 diagonals
triangles have 0 diagonals
pentagons have
5 diagonals
hexagons have 9 diagonals
A polygon with n sides has:
We use the table below to reason
inductively to get the formula for the
number of diagonals in a polygon.
n(n  3)
2
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
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