Name polygons based on their number of sides Classify polygons based on --concave or convex --equilateral, equiangular, regular Calculate and use the measures of interior and exterior angles of polygons

1. Obj. 25 Properties of Polygons
The student is able to (I can):
• Name polygons based on their number of sides
• Classify polygons based on
— concave or convex
— equilateral, equiangular, regular
• Calculate and use the measures of interior and exterior
angles of polygons

2. polygon
A closed plane figure formed by three or
more noncollinear straight lines that
intersect only at their endpoints.
polygons
not
polygons

3. vertex
The common endpoint of two sides.
Plural: vertices
vertices.
diagonal
A segment that connects any two
nonconsecutive vertices.
diagonal
regular
vertex
A polygon that is both equilateral and
equiangular.

4. Polygons are named by the number of their
sides:
Sides
Name
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n-gon

5. Examples
Identify the general name of each polygon:
1. pentagon
2. dodecagon
3. quadrilateral

6. concave
A diagonal of the polygon contains points
outside the polygon. (“caved in”)
convex
Not concave.
concave
pentagon
convex
quadrilateral

7. We know that the angles of a triangle add
up to 180º, but what about other polygons?
Draw a convex polygon of at least 4 sides:
180º
180º
180º
Now, draw all possible diagonals from one
vertex. How many triangles are there?
What is the sum of their angles?

8. Thm 6-1-1
Polygon Angle Sum Theorem
The sum of the interior angles of a
convex polygon with n sides is
(n — 2)180º
If the polygon is equiangular, then the
measure of one angle is
(n − 2)180°
n

9. Sides
Name
Triangles
Sum Int.
Each Int.
(Regular)
3
Triangle
1
(1)180º=180º
60º
4
Quadrilateral
2
(2)180º=360º
90º
5
Pentagon
3
(3)180º=540º
108º
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n-gon

10. Let’s update our table:
Sides
Name
Triangles
Sum Int.
Each Int.
(Regular)
3
Triangle
1
(1)180º=180º
60º
4
Quadrilateral
2
(2)180º=360º
90º
5
Pentagon
3
(3)180º=540º
108º
6
Hexagon
4
(4)180º=720º
120º
7
Heptagon
5
(5)180º=900º
≈128.6º
8
Octagon
6
(6)180º=1080º
135º
9
Nonagon
7
(7)180º=1260º
140º
10
Decagon
8
(8)180º=1440º
144º
12
Dodecagon
10
(10)180º=1800º
150º
n
n-gon
n—2
(n — 2)180º
(n − 2)180°
n

11. An exterior angle is an angle created by
extending the side of a polygon:
Exterior
angle
Now, consider the exterior angles of a
regular pentagon:

12. From our table, we know that each interior
angles is 108º. This means that each
exterior angle is 180 — 108 = 72º.
72º
72º
72º
108º 72º
72º
The sum of the exterior angles is therefore
5(72) = 360º. It turns out this is true for
any convex polygon, regular or not.

13. Polygon Exterior Angle Sum Theorem
The sum of the exterior angles of a
convex polygon is 360º.
For any equiangular convex polygon with
n sides, each exterior angle is
360°
n
Sides
Name
Sum Ext.
Each Ext.
3
Triangle
360º
120º
4
Quadrilateral
360º
90º
5
Pentagon
360º
72º
6
Hexagon
360º
60º
8
Octagon
360º
45º
n
n-gon
360º
360º/n