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Section 3-4
Polygon Angle-SumTheorems
•Name Polygons
•Calculate Interior Angles
•Calculate Exterior Angles
•Understand Diagonals
The Basics . . .
Definition: A polygon is a closed plane
figure with at least three sides that
are segments. The sides intersect only
at their endpoints, and no adjacent
sides are collinear.
Polygons
Not Polygons
Naming Polygons
Start at any vertex and list the vertices
consecutively in a clockwise or
counterclockwise direction.
D
E
I
N
A
DIANE
IANED
ANEDI
NEDIA
EDIAN
DENAI
ENAID
NAIDE
AIDEN
IDENA
Definition: A diagonal of a polygon is a
segment that connects two
nonconsecutive vertices.
The Basics . . .
Definition: A convex polygon has all
diagonals on the interior of the
polygon.
Definition: A concave polygon has a
diagonal on the exterior of the polygon.
Convex
Concave
Naming Polygons
# Sides
3
4
5
6
7
8
9
10
12
n
Polygon
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon
n-gon
Real-Life Connections
H
C
H
C
C
H
C
C
H
C
H
Benzene
H
Apply What You Already Know:
Theorem: The sum of the measures of
the three angles in a triangle is 180
degrees.
How About a Quadrilateral? . . .
What is the sum of the angles?
Explain your answer?
Polygons and Interior Angles
Investigation Part I
Theorem: The sum Si of the measures of
the angles of a polygon with n sides is
given by the formula: Si = (n – 2)180.
Huh?
• The polygon is divided into triangles.
• The number of triangles is always two
less than the number of polygon sides.
• There are 180 in each triangle.
Polygons and Exterior Angles
Investigation Part II
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
Huh?
exterior angle
If you cut out each
exterior angle and arranged
the vertices on top of one
another they would form a
circle of sorts.
More About Interior Angles
Investigation Part III
180n – 360 = 180(n – 2)
Huh?
If you add all the angles of each triangle,
then you include all the angles that go
completely around the selected point.
Hence, you need to subtract 360!
Diagonals of Polygons
Investigation Part III
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
Huh?
From each of the n vertices you
can draw n – 3 diagonals. Thus,
there are n(n-3) diagonals total.
But, by this method, each
diagonal is drawn twice, so you
must divide by 2.
Regular Polygons
Definition: A regular polygon is a
polygon that is both equilateral and
equiangular.
Theorem: The measure E of each
exterior angle of an equiangular polygon
of n sides is given by the formula
You already know that Se  360,
so how can you find E?
exterior angle
360
E
n