Honors Geometry
Sections 3.1 & 3.6
Polygons and Their
Angle Measures
The word polygon means many
sides.
In simple terms, a polygon is a
many-sided closed figure.
Formally, a polygon is a figure formed from
three or more line segments such that each
segment intersects exactly two other segments,
one at each endpoint, and no two segments
with a common endpoint are collinear.
The segments are called the_____
sides of
the polygon and the common
vertices of
endpoints are called the _______
the polygon.
When naming a polygon, you must
list the vertices in order either
clockwise or counterclockwise.
The polygon at the right could be
named _______
BAFEDC
ABCDEF or _______
A diagonal of a polygon is a
segment joining two nonadjacent
vertices.
A polygon is equilateral iff
all its sides are congruent.
A polygon is equiangular iff
all its angles are congruent.
A polygon that is both equilateral
and equiangular is called a
_______
regular polygon.
The center of a regular polygon is
the point which is equidistant from
each of the vertices.
Polygons are classified according to
the number of its sides.
triangle
3 - ____________
5 - ____________
pentagon
7 - ____________
heptagon
nonagon
9 - ____________
dodecagon
12 - ___________
4 - ____________
quadrilateral
hexagon
6 - ____________
octagon
8 - ____________
decagon
10 - ___________
n - ___________
n - gon
A polygon is convex iff the line
containing a side does not contain
a point in the interior.
A polygon that is not convex is
concave.
For each figure, draw all the diagonals from
one vertex and complete the table.
4
6
2
3
4
n
n2
5
2 180  360
3 180  540
4 180  720
n  2180
Theorem 3.6.1
The sum of the measures of the
interior angles of a (convex)
polygon with n sides is
n  2180
Corollary to Theorem 3.6.1
The measure of each interior angle
of a regular n-gon is
n  2180
n
Example 1: Find the sum of measures of
the interior angles of a dodecagon.
12  2180  1800
Example 2: Find the measure of each
interior angle of a regular 20-gon.
20  2180  3240  162
20
20
While the sum of the interior
angles of a polygon changes as the
number of sides changes, this is
not the case with the sum of the
exterior angles.
Theorem 3.6.3
The sum of the measures of the
exterior angles of a (convex)
polygon, one at each vertex, with n
sides is
360
Here’s an example of why that is the case.
180
180
180
180
180
Adding the five equations together, we get:
900
540
360
Corollary to Theorem 3.6.3
The measure of each exterior angle
of a regular n-gon is
360
n
Complete this table for regular polygons.
2520
22
157.5
360
22.5
163.63
360
16.36
360
(n  2)180 (n  2)180
n
(n  2)180  3600
360
360
360
n
Complete this table for regular polygons.
2520
22
157.5
360
22.5
163.63
360
16.36
4140 165.6 360
30 5040
360
25
(n  2)180 (n  2)180
n
(n  2)180
 168
n
360
12
360
n
360
 14.4
n