11.1

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11.1 Angle Measures in Polygons
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Diagonals ? Connect two nonconsecutive vertices,
and are drawn with a red dashed line.
Lets draw all the diagonals from 1 vertex.
Sides of Triangles Total degrees
3
540
5
3
Find out how many degrees are in these two
shapes, and try to make a formula
Sides of Triangles Total degrees
540
3
5
720
4
6
5
900
7
n-2
(n-2)180
n
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Remember, angles on the outside are EXTERIOR
ANGLES.
What do all the exterior angles of a octagon add
up to?
What do all the Exterior Angles of a polygon add
up to?
What do all the exterior angles of a decagon add
up to?
360 degrees!!
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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.
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What is the measure of one interior angle of a
regular pentagon?
What is the measure of one interior angle of a
regular octagon?
The general formula for the measure of one
interior angle of a REGULAR polygon is ?
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Fill out this regular polygon chart here.
Think about the relationship between interior and
exterior angles.
Interior and exterior angles are supplementary.
Sides Name Total interior Each interior Total exterior Each exterior
4
8
12
8
Sum of interior angles in polygon
Sum of exterior angles in polygon
Measure of ONE interior angle of REGULAR polygon
Measure of ONE exterior angle of REGULAR polygon
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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.
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11.2 Areas of Regular Polygons
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Area of Equilateral triangle.
s
8
13
Central Angle ? Angle formed from center of
polygon to consecutive vertices.
Radius
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.
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The area of these 5 triangles is
Or we can think of it as
What do you think we can do to find the area of
this shape?
So you see its
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Lets find the area of a pentagon with side
length 10
Which trig function do we use to find the apothem?
72o
TANGENT!
Plug in, be careful with the perimeter!
36o
5
10
16
10
10
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11.3 Perimeters and Areas of Similar Figures
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Find the perimeter and area of a rectangle with
dimensions
4 by 10
28
40
8 by 20
56
160
6 by 15
42
90
20 by 50
140
1000
2 by 5
14
10
Side Ratio Perimeter Ratio Area Ratio
15
15
125
43
43
169
31
31
91
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Do you notice a relationship between the side
ratio, perimeter ratio, and area ratio?
Theorem 11-5 If the scale factor of two similar
figures is ab, then 1) The ratio of perimeters
is ab 2) The ratio of areas is a2b2
Find the area and perimeter of a rectangle with
dimensions
4 by 10
28
40
8 by 20
56
160
6 by 15
42
60
20 by 50
140
1000
2 by 5
14
10
Side Ratio Perimeter Ratio Area Ratio
15
15
125
43
43
169
31
31
91
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Find the perimeter ratio and the area ratio of
the two similar figures given below.
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Two basic problems I have two pentagons. If the
area of the smaller pentagon is 100, and they
have a 14 side length ratio, then what is the
area of the other pentagon?
I have 2 dodecagons. If the area of one is 314
and the other is 942, what is the side length
ratio?
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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?
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11.4 Circumference and Arc Length
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Circumference is the distance around the circle.
(Like perimeter) C pd 2pr
LIKE THE CRUST PIZZA PART
Area of a circle A pr2
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A
O
x
B
Like crust
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Find the length of the arc
3
O
120o
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Find the length of the arc
5
O
100o
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Find the length of the arc
O
30
20o
30
Radius 5 6
mAB 30o 60o 135o
Length of AB 4p 9p 5p
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Find x and y
O
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Find the Perimeter of this figure.
Do not subtract and then square, must do each
circle separately!
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Find Perimeter of red region.
4
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Find the length of green part
30o
6
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11.5 Areas of Circles and Sectors
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Circumference is the distance around the circle.
(Like perimeter) C pd 2pr
LIKE THE CRUST PIZZA PART
Area of a circle A pr2
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Find the area of a circle with diameter 8 in.
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Fake sun has a radius of .5 centimeters. Find the
circumference and area of fake sun.
Circumference 2p(.5) p
Area p(.5)2 .25p
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Find the area of the shaded part.
10
8
5
6
40
A
O
x
B
Like crust Like the slice
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Find the area of the sector.
3
O
120o
42
Find the area of the sector.
4
O
90o
43
Find the area of the sector.
10
O
160o
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Find area of blue part and length of green part
30o
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