Polygons and Their Angles

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Polygons and Their Angles

• 1-6 and 6-1

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Polygon A closed figure. They have vertices,
sides, angles, and exterior angles.
You name a polygon by just listing its vertices
in order around the polygon.
ABCDEF
One name for this polygon is _____________________
____
Diagonal a segment connecting two NONconsecutive
vertices of a polygon.
AC or AD or AE
Diagonal Example __________________
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Convex Polygons - polygons where no diagonal goes
outside the figure.
Concave Polygons - polygons where any diagonal
goes outside of the figure. Concave polygons
cave in.
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Classifying Polygons

• You can classify a polygon by the number of sides
  it has. YOU WILL BE EXPECTED TO KNOW THESE!!

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INTERIOR ANGLE SUM THEOREM
The sum of the measures of the angles in a convex
polygon with n sides is (n - 2)180
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Exterior Angle Sum Theorem
The sum of the measure of the exterior angles of
ANY convex polygon, one at each vertex is 360
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Example
Find the interior and exterior angle sums for
each polygon
1. quadrilateral 2. 12-gon 3. hexagon 4.
nonagon
5. decagon 6. pentagon 7. octagon 8. 18-gon
(4-2)180 360
(10-2)180 1440 Ext. Angle Sum 360
Exterior Angle sum is always 360
(12-2)180 1800 Ext. Angle Sum 360
(5-2)180 540 Ext. Angle sum 360
(6-2)180 720 Ext. Angle Sum 360
(8-2)180 1080 Ext Angle Sum 360
(9-2)180 1260 Ext. Angle Sum 360
(18-2)180 2880 Ext Angle Sum 360
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Example 2
Find the value of x
You should get x 90
You should get x 100
Since there are 5 sides, then the interior angle
sum is (5-2)180 or 540. Then take 360 - 90 - 90 -
160 - 150 to get x. You should get 50
Do the same for the others. Count the number of
sides and figure the interior angle sum.
Then subtract out the angles that you already
know.
You know there are 4 sides, so the interior
angle sum is 360. Take 360 - 60 and you get
300. Then divide by 3 and each angle is 100.
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Regular Polygons
Regular Polygons - a polygon that is BOTH
equilateral AND equiangular
If you see the word REGULAR, it means the figure
is special and you can divide by the number of
sides to get individual angle measures
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Example 3
For each REGULAR polygon, find the measure of
each interior angle and exterior angle.
13. triangle 14. quadrilateral 15. hexagon
16. decagon 17. 15-gon
Interior (3-2)180 / 3 60
Exterior 360 / 3 120
Interior (10-2)180 / 10 144 Exterior 360 /
10 36
Interior (4-2)180 / 4 90 Exterior 360 / 4
90
Interior (15-2)180 / 15 156 Exterior 360 /
15 24
Interior (6-2)180 / 6 120 Exterior 360 / 6
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NOTICE interior and exterior angles add to
180!!!!
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Example 4
How many sides does a regular polygon have if the
measure of each exterior angle is
Just take 360 divided by each angle to get your
answer
18. 60 19. 15 20. 120
6 sides
24 sides
3 sides
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Example 5
How many sides does a regular polygon have if the
measure of each interior angle is
Since the interior and exterior angles Add to
180, find the exterior angle first!!!
Interior angle is 60, so exterior angle is
180-60 120. Now do 360 divided by 120. You
should get 3.
21. 60 22. 160 23. 144
Interior angle is 160. 180-160 20, so exterior
angle is 20. Now do 360 divided by 20. You get 18
Interior angle is 144. 180-144 36, so exterior
angle is 36. Now do 360 divided by 36. You get 10.
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Have a great day!!