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 # of Triangles Total degrees

5 3 540

Find out how many degrees are in these two shapes, and try to make a formula

Sides # of Triangles Total degrees

7 5 900

n n-2 (n-2)180

5 3 540

6 4 720

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?

5

180)25(

1085

540

What is the measure of one interior angle of a regular octagon?

8

180)28(

1358

1080

The general formula for the measure of one interior angle of a REGULAR polygon is

n

n 180)2(

Fill out this regular polygon chart here.

Sides Name Total interior

Each interior

Total exterior

Each exterior

4

8

12

Think about the relationship between interior and exterior angles.

Interior and exterior angles are supplementary.

n

360

180)2( n

n

n 180)2(

360

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

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?

Interior and exterior angles are supplementary.

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?

11.2 – Areas of Regular Polygons

4

32sA

Area of Equilateral triangle.

8s

Central Angle Angle formed from center of polygon to consecutive vertices.

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.

n

360

Radius

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 it’s

bhbhbhbhbhA2

1

2

1

2

1

2

1

2

1

PhhbA2

1)5(

2

1

apothemtheisaaPA2

1

Let’s find the area of a pentagon with side length 10

105

Which trig function do we use to find the apothem?

Plug in, be careful with the perimeter!

TANGENT!

a

536tan

72o

36o

8819.6a

)50)(8819.6(2

1A 0477.172

10

10

11.3 – Perimeters and Areas of Similar Figures

Find the perimeter and area of a rectangle with dimensions:

4 by 10

8 by 20

6 by 15

20 by 50

2 by 5

28

56

42

140

14

40

160

90

1000

10

Side Ratio Perimeter Ratio Area Ratio1:5 1:5 1:254:3 4:3 16:93:1 3:1 9:1

Find the area and perimeter of a rectangle with dimensions:

4 by 10

8 by 20

6 by 15

20 by 50

2 by 5

28

56

42

140

14

40

160

60

1000

10

Side Ratio Perimeter Ratio Area Ratio1:5 1:5 1:254:3 4:3 16:93:1 3:1 9:1

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 a:b, then:

1) The ratio of perimeters is a:b

2) The ratio of areas is a2:b2

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?

I have 2 dodecagons. If the area of one is 314 and the other is 942, what is the side length ratio?

22 4:1

16:1

x

100

16

1

1600x

3:1:

3:1

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

Area of a circle:

A = πr2

LIKE THE CRUST

PIZZA PART

.ncecircumfereofPart

lengthABofLength

DEGREESINMEASURED

arcofMeasuremAB

angletheofmeasureisx

rx

ABofLength 2360

x

A

BO

Like crust

O 120o

3

Find the length of the arc

)3(2360

120 arcofLength

63

1

2

O 100o

5

Find the length of the arc

)5(2360

100 arcofLength

1018

5

9

25

O

20o

30

Find the length of the arc

)30(2360

20 arcofLength

6018

1

3

10

Radius 5 6

mAB 30o 60o 135o

Length of AB

4π 9π 5π

O

Find x and y

Find the Perimeter of this figure.

12

20)20(2)12(288

1664

162440

Do not subtract and then square, must do each circle separately!

4

Find Perimeter of red region.

30o

6

Find the length of green part

11.5 – Areas of Circles and Sectors

Circumference is the distance around the

circle. (Like perimeter)

C = πd = 2πr

Area of a circle:

A = πr2

LIKE THE CRUST

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.

Circumference:

2π(.5) = π

Area:

π(.5)2 = .25π

6

8

Find the area of the shaded part.

2

8652

10

5

2425

.ncecircumfereofPart

lengthABofLength

DEGREESINMEASURED

arcofMeasuremAB

angletheofmeasureisx

rx

ABofLength 2360

x

angletheofmeasureisx

rx 2

360AOBsectorofArea

A

BO

Like crust Like the slice

O 120o

3

Find the area of the sector.

2)3(360

120sectorofArea

93

1

3

O 90o

4

Find the area of the sector.

2)4(360

90sectorofArea

164

1

4

O 160o

10

Find the area of the sector.

2)10(360

160sectorofArea

1009

4

9

400

30o

6

Find area of blue part and length of green part