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
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
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