Objectives:
Develop and use formulas for the sums of the
measures of interior and exterior angles of polygons
Warm-Up:
Here’s a two part puzzle
designed to prove that half
of eleven is six. First
rearrange two sticks to
reveal the number eleven.
Then remove half of the
sticks to reveal the number
six.
Convex Polygon:
A polygon in which any line segment
connecting two points of the polygon has no
part outside the polygon.
Concave Polygon:
A polygon that is not convex.
Consider the following Pentagon:
Divide the polygon into three
triangular regions by drawing all the
possible diagonals from one vertex.
Find each of the following:
𝟐
𝟑
𝟒
𝟏
𝟓
𝟗
𝟖
𝒎 < 𝟏 + 𝒎 < 𝟐 + 𝒎 < 𝟑 = ______
𝒎 < 𝟒 + 𝒎 < 𝟓 + 𝒎 < 𝟔 = ______
𝒎 < 𝟕 + 𝒎 < 𝟖 + 𝒎 < 𝟗 = ______
Add the three
expressions:
𝟕
𝟔
Note:
You can form triangular regions by drawing all possible
diagonals from a given vertex of any polygon
Polygon
# of
sides
# of
triangular
regions
Sum of
Interior
angles
Triangle
3
1
180
Quadrilateral
4
2
360
Pentagon
5
3
540
Hexagon
6
4
720
n-gon
n
n−2
180(n-2)
The sum of the measures of the interior
angles of a polygon with n sides is:
180(n-2)
Note:
Recall that a regular polygon is on in which all the
angles are congruent.
Polygon
# of
sides
Sum of
Interior
angles
Measure
of Interior
angles
Triangle
3
180
60
Quadrilateral
4
360
90
Pentagon
5
540
108
Hexagon
6
720
120
n-gon
n
180(n-2)
180(n-2)
n
The measure of an Interior Angle of a
Regular Polygon with n sides is:
180(n-2)
n
Exterior Angle Sums in Polygons
Polygon
# of
sides
Sum of
interior &
exterior
angles
Sum of
Interior
angles
Sum of
Exterior
angles
Triangle
3
540
180
360
Quadrilateral
4
720
360
360
Pentagon
5
900
540
360
Hexagon
6
1080
720
360
n-gon
n
180n
180(n-2)
360
Theorem
Sum of the measures of the Exterior
Angles of a Polygon is:
𝟑𝟔𝟎
𝟎
For a Convex Polygon
Polygon
Number Number
of
of Δ
Sides Regions
Sum of
Interior
Angles
For a Regular Polygon
Sum of Sum of Measure of
Exterior Int & Ext
Interior
Angles Angles
Angles
Measure
of
Exterior Angles
Triangle
3
1
180
360
540
60
120
Quadrilateral
4
2
360
360
720
90
90
Pentagon
5
3
540
360
900
108
72
Hexagon
6
4
720
360
1080
120
60
Heptagon
7
5
900
360
1260
128.6
51.4
Octagon
8
6
1080
360
1440
135
45
Nonagon
9
7
1260
360
1620
140
40
Decagon
10
8
1440
360
1800
144
36
11-gon
11
9
1620
360
1980
147.3
32.7
Dodecagon
12
10
1800
360
2160
150
30
13-gon
13
11
1980
360
2340
152.3
27.2
n-gon
n
n-2
180(n-2)
360
180n
𝟏𝟖𝟎(𝐧 − 𝟐)
𝐧
𝟏𝟖𝟎 −
𝟏𝟖𝟎(𝐧 − 𝟐)
𝐧
Find the indicated angle measures.
𝟏𝟏𝟓𝟎
𝟓𝟒𝟎
𝟏𝟐𝟎𝟎
𝒚𝟎
𝒙𝟎
𝟒𝟖𝟎
𝒛𝟎
𝟏𝟎𝟎𝟎
Find the indicated angle measures.
𝟗𝟎
𝟖𝟓
𝟎
𝟎
𝒚𝟎
𝟕𝟓𝟎
𝒙𝟎
𝟏𝟑𝟐𝟎
𝟏𝟐𝟎𝟎
Find the indicated angle measures.
𝟏𝟑𝟎𝟎
𝒙𝟎
𝟗𝟎𝟎
𝟏𝟎𝟎𝟎
𝟏𝟏𝟎𝟎
Find the indicated angle measures.
𝒙𝟎
𝟏𝟎𝟓𝟎
𝟏𝟏𝟎𝟎
𝟕𝟓𝟎
𝟏𝟎𝟎𝟎
For each polygon determine the measure of an interior
angle and the measure of an exterior angle.
A rectangle
An equilateral
triangle
A regular
dodecagon
An equiangular
pentagon
An interior angle measure of a regular polygon is given.
Find the number of sides of the polygon
𝟎
𝟏𝟑𝟓
𝟏𝟓𝟎
𝟎
𝟎
𝟏𝟔𝟓
An exterior angle measure of a regular polygon is given.
Find the number of sides of the polygon
𝟎
𝟔𝟎
𝟑𝟔
𝟎
𝟎
𝟐𝟒
Find the indicated angle measure.
(𝒙)𝟎
(𝟒𝒙)𝟎
(𝟑𝒙)𝟎
(𝟐𝒙)𝟎
Find the indicated angle measure.
(𝒙𝟐 +𝟐𝒙 + 𝟏𝟎)𝟎
(𝟐𝒙 + 𝟑𝟎)𝟎
(𝟖𝒙 − 𝟏𝟎)𝟎
(𝒙𝟐 +𝟏𝟎)𝟎
Find the indicated angle measure.
(𝟒𝒙 + 𝟑𝟐)𝟎
(𝟐𝒙 + 𝟒)𝟎
(𝟓𝒙 − 𝟏𝟒)𝟎
(𝟒𝒙 − 𝟑𝟕)𝟎
(𝟓𝒙 + 𝟏𝟓)𝟎