Download 11.1 – Angle Measures in Polygons

11.1 – Angle Measures in
Polygons
Diagonals  Connect two
nonconsecutive vertices, and are
drawn with a red dashed line.
Let’s draw all the
diagonals from 1
vertex.
Sides
5
# of Triangles
3
Total degrees
540
Find out how many degrees are in these
two shapes, and try to make a formula
Sides
# of Triangles
5
6
7
n
3
4
5
n-2
Total degrees
540
720
900
(n-2)180
Remember, angles on
the outside are
EXTERIOR ANGLES.
What do all the
Exterior Angles of a
polygon add up to?
360 degrees!!
What do all the
exterior angles
of a octagon
add up to?
What do all the
exterior angles of a
decagon add up to?
Theorem 11-1 (Sum of interior angles of
polygon)  The sum of the measures of
the angles of a convex polygon with n
sides is (n-2)180
Theorem 11-2 (Exterior angles sum
theorem)  The sum of the measure of
the exterior angles of a convex polygon
is 360.
What is the measure of
one interior angle of a
regular pentagon?
What is the measure of
one interior angle of a
regular octagon?
(5  2)  180
5
(8  2)  180
8
540
 108
5
1080
 135
8
The general formula for
the measure of one
interior angle of a
REGULAR polygon is 
( n  2)  180
n
Fill out this regular polygon chart here.
Think about the relationship between interior and exterior
angles.
Interior and exterior angles are supplementary.
Sides
4
8
12
Name
Total
interior
Each
interior
Total
Each
exterior exterior
Sum of interior angles in
polygon
Sum of exterior angles in
polygon
(n  2)  180
360
Measure of ONE interior
angle of REGULAR
polygon
Measure of ONE exterior
angle of REGULAR
polygon
( n  2)  180
n
360
n
How many sides are there if
the one interior angle of a
regular polygon is 135
degrees?
How many sides are there if
the one exterior angle of a
regular polygon is 45 degrees?
How many sides are there if
the one interior angle of a
regular polygon is 170
degrees?
How many sides are there if
the one exterior angle of a
regular polygon is 20 degrees?
Interior and exterior angles are supplementary.
11.2 – Areas of Regular
Polygons
Area of Equilateral triangle.
s2 3
A
4
s
8
Central Angle  Angle
formed from center of
polygon to consecutive
360
vertices.

n
Apothem  Distance
from center of polygon to
side.
Things to notice, all parts
can be found using
SOHCAHTOA.
It is isosceles, so you
can break up the triangle
in half.
Radius
The area of these 5
triangles is =
1
1
1
1
1
A  bh  bh  bh  bh  bh
2
2
2
2
2
1
1
Or we can think of it as
A  (5b ) h  Ph
2
2
What do you think we can do to find the area of this
shape?
1
So you see it’s A  aP a is the apothem
2
Let’s find the area of a
pentagon with side length 10
Which trig function do we
use to find the apothem?
TANGENT!
5
tan 36 
a
72o
a  6.8819
Plug in, be careful with the
perimeter!
1
A  (6.8819)(50)  172.0477
2
36o
5
10
10
10
11.3 – Perimeters and Areas
of Similar Figures
Find the perimeter and area of a
rectangle with dimensions:
4 by 10
28
40
8 by 20
6 by 15
56
42
160
90
20 by 50
2 by 5
140
14
1000
10
Side Ratio
1:5
4:3
3:1
Perimeter Ratio
1:5
4:3
3:1
Area Ratio
1:25
16:9
9:1
Do you notice a relationship between the side ratio,
perimeter ratio, and area ratio? Theorem 11-5
Find the area and perimeter of a
If the scale factor of
rectangle with dimensions:
two similar figures is
4 by 10
28
40
a:b, then:
8 by 20
56
160
1) The ratio of
6 by 15
42
60
perimeters is a:b
20 by 50
140
1000
2) The ratio of
2 by 5
14
10
areas is a2:b2
Side Ratio
1:5
4:3
3:1
Perimeter Ratio
1:5
4:3
3:1
Area Ratio
1:25
16:9
9:1
Find the perimeter ratio and the area ratio of the two similar
figures given below.
Two basic problems:
I have two pentagons.
If the area of the
smaller pentagon is
100, and they have a
1:4 side length ratio,
then what is the area
of the other pentagon?
12 : 4 2
1:16
x  1600
1 100

16
x
I have 2 dodecagons.
If the area of one is
314 and the other is
942, what is the side
length ratio?
1: 3
1: 3
Two basic problems:
A cracker has a perimeter
of 10 inches. A similar
mini cracker has
perimeter 5 inches. If the
area of the regular
cracker is 20 in2, what is
the area of the mini
cracker?
I have 2 n-gons. If the
area of one is 135 and
the other is 16, what is
the perimeter ratio?
11.4 – Circumference and Arc
Length
Circumference is the
distance around the
circle. (Like perimeter)
C = πd = 2πr
LIKE THE CRUST
Area of a circle:
A = πr2
PIZZA PART
mAB  Measure of arc
MEASURED IN DEGREES
Length of AB  length
Part of circumference.
x
Length of AB 
2 r
360
x is measure of the angle
Like crust
A
O x
B
Find the length of the arc
120
Length of arc 
2 (3)
360
1
 6
3
 2
3
O 120o
Find the length of the arc
100
Length of arc 
2 (5)
360
5
 10
18
25
 
9
5
O 100o
Find the length of the arc
20
Length of arc 
2 (30)
360
1
 60
18
10
 
3
O
30
20o
Radius
5
mAB
30o
Length
of AB
6
60o
4π
135o
9π
5π
Find x and y
O
Find the Perimeter of
this figure.
12
2 ( 20 )
2 (12 )
20
88
40  24  16
64  16
Do not subtract and then square,
must do each circle separately!
Find Perimeter of red region.
4
Find the length of green part
30o
6
11.5 – Areas of Circles and
Sectors
Circumference is the
distance around the
circle. (Like perimeter)
C = πd = 2πr
LIKE THE CRUST
Area of a circle:
A = πr2
PIZZA PART
Find the area of a circle with diameter 8 in.
Fake sun has a radius
of .5 centimeters.
Find the circumference
and area of fake sun.
Area:
Circumference:
2π(.5) = π
π(.5)2 = .25π
Find the area of the
shaded part.
10
8
5
6
68
5 
2
25  24
2
mAB  Measure of arc
MEASURED IN DEGREES
Length of AB  length
Part of circumference.
A
O x
B
x
Length of AB 
2 r Area of sector AOB  x  r 2
360
360
x is measure of the angle x is measure of the angle
Like crust
Like the slice
Find the area of the sector.
120
Area of sector 
 (3) 2
360
1
 9
3
 3
3
O 120o
Find the area of the sector.
90
Area of sector 
 (4) 2
360
1
 16
4
 4
4
O 90o
Find the area of the sector.
160
Area of sector 
 (10) 2
360
4
 100
9
400


9
10
O 160o
Find area of blue part and length of green part
30o
6