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Unit 7
Polygons
Lesson 7.1
Interior & Exterior
Angle Sums
of Polygons
Lesson 7.1 Objectives
• Calculate the sum of the interior
angles of a polygon. (G1.5.2)
• Calculate the sum of the exterior
angles of a polygon. (G1.5.2)
• Classify different types of
polygons.
Definition of a Polygon
•
1.
2.
3.
4.
5.
A polygon is plane figure (twodimensional) that meets the
following conditions.
It is formed by three or more segments called sides.
The sides must be straight lines.
Each side intersects exactly two other sides, one at each
endpoint.
The polygon is closed in all the way around with no
gaps.
Each side must end when the next side begins. No tails.
Polygons
Not Polygons
No Curves
Too Many
Intersections
No Gaps
Types of Polygons
Number of Sides
Type of Polygon
3
Triangle
4
Quadrilateral
Pentagon
Hexagon
Heptagon
5
6
7
9
Octagon
Nonagon
10
Decagon
12
Dodecagon
n
n-gon
8
Concave v Convex
•
•
A polygon is convex if
no line that contains a
side of the polygon
contains a point in the
interior of the
polygon.
Take any two points
in the interior of the
polygon. If you can
draw a line between
the two points that
never leave the
interior of the
polygon, then it is
convex.
•
•
•
A polygon is concave if
a line that contains a
side of the polygon
contains a point in the
interior of the
polygon.
Take any two points in
the interior of the
polygon. If you can
draw a line between
the two points that
does leave the interior
of the polygon, then it
is concave.
Concave polygons
have dents in the
sides, or you could say
it caves in.
Example 7.1
Determine if the following are polygons or not.
If it is a polygon, classify it as concave or convex and
name it based on the number of sides.
1.
4.
Yes
No!
2.
5.
Yes
Concave
Octagon
No!
Concave
Hexagon
6.
3.
Yes
Yes
Convex
Pentagon
Concave
Heptagon
Diagonals of a Polygon
• A diagonal of a polygon is a
segment that joins two
nonconsecutive vertices.
– A diagonal does not go to the
point next to it.
• That would make it a side!
– Diagonals cut across the
polygon to all points on the
other side.
• There is typically more than one
diagonal.
Interior Angles of a Polygon
• The sum of the interior angles of a triangle is
– 180o
• The sum of the interior angles of a
quadrilateral is
180o
– 360o
• The sum of the interior angles of a pentagon is
– ???
360o
• The sum of the interior angles of a hexagon is
– ???
• By splitting the interior into triangles, it should
be able to tell you the sum of the interior
angles.
– Pick one vertex and draw all possible diagonals
from that vertex.
– Then, count up the number of triangles and
multiply by 180o.
540o
720o
Theorem 11.1:
Polygon Interior Angles Theorem
• The sum of the measure of the
interior angles of a convex
n-gon is
180o (n  2)
n = number of sides
Example 7.2
Find the sum of the interior angles of the
following convex polygons.
1.
180o (n  2)
n5
2.
180o (5  2)
180o (3)  540o
180o (n  2)
n  12
180o (12  2)
180o (10)  1800o
3.
nonagon 180o (n  2)
n9
4.
17-gon
n  17
180o (9  2)
180o (7)  1260o
180o (n  2)
180o (17  2)
180o (15)  2700 o
Example 7.3
Find x.
1.
180o (6  2)
720 o
2.
180o (8  2)
1080o
3.
180o (8  2)
1080o
720  88 105 142 140 124  x
720  599  x
x  121
1080  824  32x
256  32x
x 8
1080  824  32x
256  32x
x 8
Exterior Angles
• An exterior angle is formed by
extending each side of a polygon
in one direction.
– Make sure they all extend either
pointing clockwise or counterclockwise.
4
3
5
2
1
Theorem 11.2:
Polygon Exterior Angles Theorem
• The sum of the measures of the
exterior angles of a convex
polygon is 360o.
– As if you were traveling in a circle!
4
3
5
2
1
 1 +  2 +  3 +  4 +  5 = 360o
Example 7.4
Find the sum of the exterior angles of the
following convex polygons.
1. Triangle
1.
2.
Quadrilateral
2.
3.
3600
Dodecagon
6.
7.
3600
Heptagon
5.
6.
3600
Hexagon
4.
5.
3600
Pentagon
3.
4.
3600
3600
17-gon
7.
3600
Example 7.5
Find x.
1.
2.
3.
Lesson 7.1 Homework
• Lesson 7.1 – Interior & Exterior
Angle Sums of Polygons
• Due Tomorrow
Lesson 7.2
Each Interior & Exterior
Angle of a
Regular Polygon
Lesson 7.2 Objectives
• Calculate the measure of each interior
angle of a regular polygon. (G1.5.2)
• Calculate the measure of each interior
angle of a regular polygon. (G1.5.2)
• Determine the number of sides of a
regular polygon based on the measure of
one interior angle.
• Determine the number of sides of a
regular polygon based on the measure of
one exterior angle.
Regular Polygons
• A polygon is equilateral if all of its sides
are congruent.
• A polygon is equiangular if all of its interior
angles are congruent.
• A polygon is regular if it is both equilateral
and equiangular.
Remember: EVERY side must be marked with the same
congruence marks and EVERY angle must be marked with the
same congruence arcs.
Example 7.6
Classify the following polygons as equilateral,
equiangular, regular, or neither.
4.
1.
2.
5.
3.
6.
Corollary to Theorem 11.1
• The measure of each interior
angle of a regular n-gon is
found using the following:
Sum of the
Interior Angles
180o (n  2)
n
Divided equally
into n angles.
It basically says to take the sum of the
interior angles and divide by the number of
sides to figure out how big each angle is.
Example 7.7
Find the measure of each interior angle in the
regular polygons.
1. pentagon
2.
decagon
3.
17-gon
Finding the Number of Sides
• By knowing the measurement of one interior
angle of a regular polygon, we can determine
the number of sides of the polygon as well.
• How?
– Since we know that all angles are going to have
the same measure we will multiply the known
angle by the number of sides of the
polygon.
– That will tell us how many sides it would take to
be set equal to the sum of all the interior
angles of the polygon.
• However, since we do not know the number of
sides of the polygon, nor do we know the total
sum of the interior angles of that polygon we
are left with the following formula to work
with:
( Angle Measure) ( Number of Sides)  ( Sum of the Interior Angles)
( Angle Measure) (n)  180o (n  2)
Example 7.8
Determine the number of sides of the regular
polygon given one interior angle.
1. 120o
2.
140o
3.
147.27o
Corollary to Theorem 11.2
•
Review: What is the
sum of the exterior
angles of a pentagon?
•
•
3600
any polygon?
•
•
3600
dodecagon?
•
•
3600
hepatagon?
•
•
•
•
This can also be
worked in “reverse” to
determine the number
of sides of a regular
polygon given the
measure of an exterior
angle.
How?
– Figure out how many
times that angle
measure would go
into 360o.
3600
Then how would we find
the measure of an
exterior angle if it were a
regular polygon?
–
• Say each exterior
angle is 1200. How
many exterior
angles would it
take to get to the
total for the
exterior angles?
Divide 360o by the
number of exterior
angles formed.
•
Which happens to be
the same as the
number of sides (n).
Each Exterior Angle Measure 
360
n
o
» 360120
» 3
• So n = 3 
360o
n
Exterior Angle Measure
Example 7.9
Find the measure of each exterior angle of the
regular polygon.
1. octagon
2.
dodecagon
3.
15-gon
Example 7.10
Determine the number of sides of the regular
polygon given the measure of an exterior
angle.
1. 72o
2.
36o
3.
27.69o
Lesson 7.2 Homework
• Lesson 7.2 - Each Interior & Exterior
Angle of a Regular Polygon
• Due Tomorrow
Lesson 7.3
Day 1:
Area and Perimeter of
Regular Polygons
Lesson 7.3 Objectives
• Calculate the measure of the central
angle of a regular polygon.
• Identify an apothem
• Calculate the perimeter and area of a
regular hexagon using equilateral
triangles. (G1.5.1)
• Calculate the perimeter and area of a
regular polygon. (G1.5.2)
• Utilize trigonometry to find missing
measurements in a regular polygon.
Parts of a Polygon
• The center of a polygon is the center
of the polygon’s circumscribed circle.
– A circumscribed circle is one in that is
drawn to go through all the vertices of a
polygon.
• The radius of a polygon is the radius
of its circumscribed circle.
– Will go from the center to a vertex.
r
Central Angle of a Polygon
• The central angle of a polygon is the
angle formed by drawing lines from
the center to two consecutive vertices.
•
This is found in a regular polygon by:
360o
Central Angle (CA) 
n
• That is because the total degrees traveled around the
center would be like a circle.
• Then divide that by the number of sides because that
determines how many central angles could be formed.
Example 7.11
Find the central angle of the following
regular polygons.
1. pentagon
2.
heptagon
3.
decagon
4.
18-gon
Reminder of - Postulate 22:
Area of a Square Postulate
• The area of a square is the square of the
length of its side.
A s
s
2
Theorem 11.3:
Area of an Equilateral Triangle
• Area of an equilateral triangle is:
AEquilateral Triangle
s
 s2 
  3
4
Example 7.12
Find the area of the equilateral triangles.
1.
2.
3.
Interior Triangles of a Hexagon
• A regular hexagon is unique in that it is the
only polygon whose central angles form
vertices of interior triangles that are all
equilateral.
– Remember, an equilateral triangle is also
regular!
4
5
3
6
2
60 o
60 o
1
60 o
• So to find the area of a regular
hexagon would be like finding the
area of SIX equilateral triangles!
AHexagon
 s2 
  3
4
6
Remember that every
side of the interior
triangles form a
radius of the circle
around the vertices.
Example 7.13
Find the area of the regular hexagons.
1.
2.
3.
Lesson 7.3a Homework
• Lesson 7.3 – Area & Perimeter of Regular
Polygons (Day 1)
• Due Tomorrow
Lesson 7.3
Day 2:
Area and Perimeter of
Regular Polygons
(Using Special Triangles & Trigonometry)
Perimeter of a Regular Polygon
• Recall that the perimeter is the sum of the
lengths of all the sides of a figure.
• Well what is true about the side lengths of a
regular polygon?
– They are all equilateral.
• So the quickest and best way to find the
perimeter when all sides are congruent is:
s = side length
Pn s
n = number of sides
Do the Equilateral Triangles Still Exist?
•
What is true about equilateral triangles?
– All sides are congruent, and…
– All angles are congruent.
• And each angle must be 60o
•
What is the central angle of a regular pentagon?
• 72o
– Would that central angle help to form an equilateral
triangle?
–
•
No, because all angles must be 60o.
What is the central angle of a regular heptagon?
• 51.43o
– Would that central angle help to form an equilateral
triangle?
–
No, because all angles must be 60o.
• So the equilateral triangles are only formed in
hexagons.
– Therefore, there must be another way to find the area
of other regular polygons.
Apothem
• The apothem is the length of a line
segment in a regular polygon drawn:
1.
2.
3.
From the center of the polygon to one of
its sides.
Such that it is perpendicular to the side.
And it bisects the side of the polygon.
a
Theorem 11.4: Area of a Regular Polygon
• The area of a regular polygon is
found using:
P = perimeter
A
1
aP
2
a = apothem
P = n•s
1
A  a( n s )
2
And how do we
find the
perimeter?
s = side length
n = number of sides
Example 7.14
Find the perimeter and area of the regular polygons.
1.
2.
3.
Lesson 7.3b Homework
• Lesson 7.3 – Area & Perimeter of Regular
Polygons (Day 2)
• Due Tomorrow
Lesson 7.3
Day 3:
Area and Perimeter of
Regular Polygons
(Using Special Triangles & Trigonometry - Again)
The Apothem and the Central Angle
• Remember it is necessary to know the length
of the apothem when finding the area of a
regular polygon.
• A = 1/2a•n•s
• So what would happen if the length of the
apothem was unknown?
• Hint: Draw the central angle and what do you
see?
–
Because the apothem is a perpendicular
bisector to the side of known length
• It divides the side in half, and
• It divides the central angle in half.
oo
o 30
3060
a
Finding the Area with Only a Known Side Length
•
•
To find the area of a regular polygon with
only a known side length, you must also
know the length of the apothem.
To do so, create a small right triangle using:
1.
2.
3.
•
The apothem.
Half of the central angle.
Half of the given side length.
And then use trigonometry to solve for the
unknown apothem.
•
SOH CAH TOA
tan 36o  .7265 
36
oo
o 36
3672
a
o
a
6
a
6
.7265
a  8.26
6
Half the given side length
Example 7.15
Find the area of the regular polygons.
1.
2.
Finding the Area with Only a Known Apothem
•
•
To find the area of a regular polygon with
only a known apothem, you must also know
the side length.
To do so, create a small right triangle using:
1.
2.
3.
•
The apothem.
Half of the central angle.
Half of the given side length.
And then use trigonometry to solve for the
unknown side length.
•
SOH CAH TOA
tan 30o  .5773 
x
4.5
x  (4.5)(.5773)
x  2.60
30o
oo
o 30
3060
4.5
s  5.20
x
Don’t forget that
you just found
half the side length.
So DOUBLE it!
Example 7.16
Find the area of the regular polygons.
1.
2.
Lesson 7.3b Homework
• Lesson 7.3 – Area & Perimeter of Regular
Polygons (Day 3)
• Due Tomorrow