Lesson 6.1
Definition
A polygon is a closed plane figure formed by three or more line
segments, i.e. with at least three sides.
Pentagon
Quadrilateral
Octagon
Classifying
Polygons are classified by the number of sides .
# of sides
Name
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
10
Decagon
n
n- gon
Example
19-gon
Parts of Polygons
The line segments forming the polygon are called sides.
The points where the line segments meet are called vertices.
Diagonals are line segments connecting 2 nonconsecutive vertices.
Angles of Polygons
Interior angles are the angles inside the polygon.
Exterior angles are formed by extending the sides.
Interior and exterior angles form supplementary angles; they add
up to 180⁰.
Polygons and Triangles
The triangle is the most important and sturdiest of all polygons.
Any polygon can be split up into triangles.
The sum of the interior angles of a polygon.
Polygon
# of
sides
# of
‘s
Sum of the angles
Quadrilateral
4
2
180⁰ x 2 = 360 ⁰
Pentagon
5
3
180⁰ x 3 = 540 ⁰
Hexagon
6
4
180⁰ x 4 = 720 ⁰
Octagon
8
6
180⁰ x 6 = 1080 ⁰
n-gon
n
n-2
180⁰(n-2)
The sum of the angles
in a triangle is 180⁰.
Polygons and Angle Properties
Theorem 6.1-Polygon Interior angle sum theorem -The sum of
the interior angles in a polygon is 180⁰(n-2) where n is the number
of sides.
Theorem 6.2-Polygon exterior angle sum theorem- The sum of
the exterior angles in a polygon is 360⁰.
A regular polygon has congruent interior angles and congruent sides.
The measure of one angle in a regular polygon is 180⁰(n-2) .
n
Example Exercises
Solve for the variables.
Sum of the angles in pentagon = (n – 2)180 = 540 .
x = 540 – 3(90) – 127
x = 143°
Example Exercises
Solve for the variables.
Sum of the angles in quadrilateral = (n – 2)180 = 360
.
x = 360 – 120 – 118 – 65
x = 57°
Example Exercises
Solve for the variables.
Sum of the angles in pentagon = (n – 2)180 = 540 .
x = 540 – 86 – 118 – 129 – 82
x = 125°
Example Exercises
Solve for the variables.
Sum of the angles in hexagon = (n – 2)180 = 720 .
x = 720 – 120 – 118 – 117 – 121 – 122
x = 122°
Example Exercises
Solve for the variables.
Sum of the angles in pentagon = (n – 2)180 = 540 .
x = 540 – 100 – 80 – 148 – 81
x = 131°
Example Exercises
Solve for the variables.
Sum of the angles in heptagon = (n – 2)180 = 900 .
x = 900 – 2(110) – 131 – 130 – 140 – 120
x = 159°
Example Exercises
Find each measure.
Sum of the angles in hexagon = (n – 2)180 = 720 .
Interior angle of a regular hexagon 720 / 6 = 120°
Example Exercises
Find each measure.
Sum of the angles in decagon = (n – 2)180 = 1440 .
Interior angle of a regular hexagon 1440 / 10 = 144°
Exterior angle of a regular hexagon 180 – 144° = 36°
OR
Sum of the exterior angles in any polygon 360°
Sum of the exterior angles of a regular decagon 360°/ 10 = 36°
Example Exercises
Find each measure.
Sum of the angles in decagon = (n – 2)180 = 1440 .
Interior angle of a regular hexagon 1440 / 10 = 144°
Example Exercises
Find each measure.
Sum of the exterior angles in any polygon 360°
Sum of the exterior angles of a regular hexagon 360°/ 6 = 60°
Example Exercises
Find each angle measure.
Sum of the angles in octagon = (n – 2)180 = 1080 .
 Interior angle of a regular octagon 1080 / 8 = 135°
Exterior angle of a regular hexagon 360 / 8 = 45°
 2 x 45° = 90°
Example Exercises
Find each angle measure.
 45°.
 90°
 45°
Example Exercises
Find each angle measure.
Sum of the angles in nonagon = (n – 2)180 = 1260 .
 &  Exterior angle of a regular hexagon 360 / 9 = 40°
 100°