# EXPLORYNG POLYGONS AND AREA

```EXPLORYNG POLYGONS
AND AREA
Objectives:
In this chapter you will:
• Find the measures of interior and exterior angles
of polygons
• Find areas of polygons and circles
• Solving problems involving geometric probability
• Determine characteristics of networks and
• Solve problems from real world
1
GEOMETRY THEN AND NOW
• Polygons have always been used in structures. In the
1400s, the first structure that resembled a roller coaster
was built in Russia. Its supporting framework was made
of triangles and parallelograms, much like today’s
coasters. Interlocking polygons make up the structure of
modern geodesic domes like Spaceship Earth at Walt
Disney World’s EPCOT center. This spherical shell is 18
stories tall.
• How are polygons used in the designs of buildings in
2
Polygons
Are polygons useful in our world? Do
you see any need to understand what
polygons are or how the knowledge of
through life? Take a look around your
world…can you really imagine a world
without polygons.
3
4
Did you know?
• Seventeen-year-old (then) Ryan Morgan
of Baltimore, Maryland, has a geometry
theorem named after him. As a high
school freshman, Ryan investigated a
theorem called (Marino)Walter’s Theorem.
This theorem states that if you divide the
sides of a triangle into equal thirds and
draw lines from each division point to the
opposite vertex, the lines form a hexagon
inside the triangle.
5
Did you know?
• (Mathematics Teacher 1993, Maushard
1994, Morgan 1994)
• Furthermore, the area of the hexagon is
one tenth of the triangle’s area. Ryan
began experimenting with dividing the
sides of triangles by other numbers. Ryan
eventually found that dividing a triangle’s
sides by any odd number and connecting
the endpoints of each middle segment to
the opposite vertex also forms a hexagon. 6
Did you know?
• The area of the hexagon is always
proportional to the area of the original
triangle. Once his conjecture was proved
by experts, it was officially named “
“Morgan’s Theorem” in his honor.
7
8
11.1 Angle Measures in
Polygons
NCSCOS: 2.02; 2.03
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Essential Question:
• How can we use measures of angles of
polygons to solve real-life problems?
• How do we find the measures of
interior and exterior angles of
polygons?
10
Special types of Polygons
• Convex- no line that contains a side of a
polygon goes through its interior
• Concave- opposite of a convex
• Equilateral- all sides are 
• Equiangular- all angles are 
• Regular polygon- equilateral and
equiangular
Measures of Interior and Exterior Angles
• In lesson 6.1, you found the sum of the
measures of the interior angles of a quadrilateral
by dividing the quadrilateral into two triangles.
You can use this triangle method to find the sum
of the measures of the interior angles of any
convex polygon with n sides, called an n-gon.
• (Okay – n-gon means any number of sides –
including 11—any given number (n).
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If all the s in a Δ add up to 180o
o
up to 360 , what about a pentagon?
2 * 180 = 3600
3 * 180 = 540o
4 * 180 = 720o
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Measures of Interior and Exterior
Angles
• For instance . . . Complete this table
Polygon
Triangle
# of
sides
3
# of
triangles
1
Sum of measures of
interior ’s
1●180=180
2●180=360
Pentagon
Hexagon
Nonagon (9)
n-gon
n
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Polygon Interior Angles
Theorem
• The sum of the
measures of the
interior angles of a
convex n-gon is
(n – 2) ● 180
• COROLLARY:
The measure of each
interior angle of a
regular n-gon is:
1
n
or
● (n-2) ● 180
( n  2)(180)
n
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Ex: What is the sum of the
measures of the interior s of a
dodecagon?
First, how many sides does a
dodecagon have?
n = 12
180(12-2) =
180(10) =
1800o
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Ex: Find the value of x.
114o
105o
Sum of all s is 540o
114+105+102+135=456o
540 – 456 = 84o
So, x = 84
102o
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Ex: The measure of each interior
of a regular polygon is 165o. How
many sides does the polygon have?
180(n  2)
 165
n
180(n-2) = 165n
180n-360 = 165n
-360 = -15n
24 = n
24 sides
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Notes
• The diagrams on the next slide show that
the sum of the measures of the exterior
angles of any convex polygon is 360.
You can also find the measure of each
exterior angle of a REGULAR polygon.
19
Copy the item below.
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Thm 11.2 – Polygon Exterior s
thm
The sum of the measures of the
exterior s of a convex polygon,
one at each vertex, is 360o.
2
m1 + m2 + m3 +
m4 + m5 = 360o
1
3
5
4
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Ex: Solve for y.
y
2y
2y
y
2y + y + 2y + y = 360
6y = 360
y = 60o
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Corollary to thm 11.2
• The measure of each exterior  of
a regular n-gon is 360
n
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Ex: Solve for x.
xo
n=6
360/n = x
360/6 = x
60o = x
This is a regular hexagon.
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Using Angle Measures in Real Life
Ex.: Finding Angle measures of a polygon
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Using Angle Measures in Real Life
Ex.: Using Angle Measures of a Regular
Polygon
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Using Angle Measures in Real Life
Ex.: Using Angle Measures of a Regular
Polygon
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Using Angle Measures in Real Life
Ex. : Using Angle Measures of a Regular
Polygon
Sports Equipment: If you were designing
the home plate marker for some new type
of ball game, would it be possible to make
a home plate marker that is a regular
polygon with each interior angle having a
measure of:
a. 135°?
b. 145°?
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Using Angle Measures in Real Life
Ex. : Finding Angle measures of a polygon
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Summarizer
• Explain in words how to find the measure of
each interior angle and each exterior angle in a
regular polygon.
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