Objective:
After studying this section, you will be
able to use some important formulas
that apply to polygons.
A polygon with 3 sides can be called a 3-gon, a seven
sided polygon can be called a 7-gon. Many have
special names.
Number of Sides
(or vertices)
3
4
5
6
7
8
9
10
12
15
n
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
Pentadecagon
N-gon
How do we find the sum of all the
angles in a polygon with n sides?
To answer this question, let’s use what we know.
We know that the sum of all the angles in a 3
sided polygon is 180.
Draw a 5-sided polygon in your notes.
Pick one vertex and draw all the diagonals from
that vertex that you can.
How many triangles did you
make?
Since we know that the sum of
all the angles in a triangle
equals 180 and there are three
triangles in a pentagon,
multiply 3(180) and you will
have the sum of all the angles.
Try finding the sum of the angles in a
heptagon.
The sum Si of the measures of the angles of a
polygon with n sides is given by the formula
Si = 180(n – 2)
From time to time we may refer to the angles of
a polygon as the interior angles of the polygon.
A
1
5
B
E
2
4
D
3
C
In the diagram exterior
angles have been
formed by extending
the side of the polygon
at each vertex.
At vertex A, m1  mEAB  180 . We can add each
exterior angle to its adjacent angle, getting a sum of 180 at
each vertex. Since there are five vertices we can calculate
the total sum as 5(180) = 900.
With the sum of the interior angles in a pentagon being
3(180) or 540, if we subtract that from 900 we will have the
sum of the exterior angles of a pentagon.
The sum of the measures of the exterior angles of
a pentagon is 900 – 540 = 360.
Find the sum of the measures of the exterior
angles in a hexagon.
Theorem
If one exterior angle is taken at each
vertex, the sum Se of the measures of
the exterior angles of a polygon is
given by the formula Se = 360.
Theorem
The number d of diagonals that can be
drawn in a polygon of n sides is given by
the formula
n( n  3)
d
2
Example 1
Find the sum of the measures of the
interior angles of the figure.
Example 2
Find the number of diagonals that can
be drawn in a pentadecagon.
Example 3
What is the name of the polygon if the
sum of the measures of the angles is
1080?
Summary
Explain in your own words what
the formula Si = 180(n – 2) means.
What specifically does (n – 2)
represent?
Homework
Worksheet 7.3