The Polygon Angle-Sum
Theorems
Objectives:
a) To classify Polygons
b) To find the sums of the measures of the interior & exterior s of Polygons.
Polygon:
• A closed plane figure.
• w/ at least 3 sides (segments)
• The sides only intersect at their endpoints
• Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.
Which of the following figures are polygons?
yes No No
Example 1: Name the 3 polygons
S T
U
V W
X
Top
XSTU
Bottom
WVUX
Big
STUVWX
I. Classify Polygons by the number of sides it has.
Sides
3
4
5
6
7
8
9
10
12
n
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
N-gon
Interior Sum
II. Also classify polygons by their Shape
a) Convex Polygon – Has no diagonal w/ points
outside the polygon.
E A
B
C
D
b) Concave Polygon – Has at least one diagonal w/
points outside the polygon.
* All polygons are convex
unless stated otherwise.
III. Polygon Interior sum
4 sides
2 Δs
2 • 180 = 360
5 sides
3 Δs
3 • 180 = 540
6 sides
4 Δs
4 • 180 = 720
• All interior sums are
multiple of 180°
Th(3-9) Polygon Angle – Sum Thm
Sum of Interior
# of sides
S = (n -2) 180
Examples 2 & 3:
• Find the interior sum of a 15 – gon.
S = (n – 2)180
S = (15 – 2)180
S = (13)180
S = 2340
• Find the number of sides of a polygon if it has an sum of 900°.
S = (n – 2)180
900 = (n – 2)180
5 = n – 2
n = 7 sides
Special Polygons:
• Equilateral Polygon – All sides are .
• Equiangular Polygon – All s are .
• Regular Polygon – Both Equilateral & Equiangular.
IV. Exterior s of a polygon.
1
2 3 1
2
3
4 5
Polygon Exterior -Sum Thm • The sum of the measures of
the exterior s of a polygon is 360°.
» Only one exterior per vertex.
360
n
1
2
3
4 5 m1 + m2 + m3 + m4 + m5 = 360
For Regular Polygons
= measure of one
exterior
The interior & the exterior
are Supplementary.
Int + Ext = 180
Example 4:
• How many sides does a polygon have if it has an exterior measure of 36°.
= 36
360 = 36n
10 = n
Example 5:
• Find the sum of the interior s of a polygon if it has one exterior measure of 24.
360
n
= 24
n = 15
S = (n - 2)180
= (15 – 2)180
= (13)180
= 2340
Example 6:
• Solve for x in the following example.
x
100
4 sides
Total sum of interior s = 360
90 + 90 + 100 + x = 360
280 + x = 360
x = 80
Example 7:
• Find the measure of one interior of a regular hexagon.
S = (n – 2)180
= (6 – 2)180
= (4)180
= 720
720
6
= 120