POLYGONS and AREA Classifying Polygons Angles in Polygons Area of Squares and Rectangles Area of Triangles Area of Parallelograms Area of Trapezoids Circumference

POLYGONS and AREA

Classifying Polygons

Angles in Polygons

Area of Squares and Rectangles

Area of Triangles

Area of Parallelograms

Area of Trapezoids

Circumference and Area of Circles

OPENERS

Assignments

Reviews

POLYGONS and AREAClassifying Polygons

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POLYGON BASICS

When people use the word “SHAPE” they are usually referring to a POLYGON.

So, what is a POLYGON?

Basically, it is a CLOSED shape with “straight” sides


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CLOSED means the shape is complete

closed

NOT closed


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Straight Sides No curves

“Straight”

Curved


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What do all these shapes

have in common?

They are all simple

polygons.To be a polygon you need 2 things:

CLOSED STRAIGHT


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Why do we need to know about polygons?

Polygons show up all over in nature, science, engineering…


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http://upload.wikimedia.org/wikipedia/commons/7/7f/Giants_causeway_closeup.jpg

http://www.google.com/imgres?imgurl=http://itech.dickinson.edu/chemistry/wp-content/uploads/2008/04/heroin-2d-skeletal.png&imgrefurl=http://itech.dickinson.edu/chemistry/?tag=kiril-popov&usg=__OtRg-sm8Q12UomTLft5LY4sHgbg=&h=947&w=1100&sz=61&hl=en&start=12&itbs=1&tbnid=nrV4Tq6VALJJ3M:&tbnh=129&tbnw=150&prev=/images?q=chemical+compound&hl=en&sa=G&as_st=y&tbs=isch:1,itp:clipart

http://www.google.com/imgres?imgurl=http://johnochwat.files.wordpress.com/2008/08/stop_sign1.jpg&imgrefurl=http://johnochwat.wordpress.com/2008/08/28/in-praise-of-the-gyratory-circus/&usg=__u18GuVRwaso0oAxm2e3Jwh4gf8I=&h=600&w=400&sz=72&hl=en&start=6&itbs=1&tbnid=BWjzLQihdLCLVM:&tbnh=135&tbnw=90&prev=/images?q=stop+sign&hl=en&sa=G&as_st=y&tbs=isch:1

http://www.google.com/imgres?imgurl=http://www.istockphoto.com/file_thumbview_approve/5627739/2/istockphoto_5627739-ferris-wheel.jpg&imgrefurl=http://www.istockphoto.com/stock-photo-5627739-ferris-wheel.php&usg=__QGa_IS3uvsXSNnoeIEnd-UtWHCA=&h=380&w=380&sz=141&hl=en&start=38&itbs=1&tbnid=-MZ5_ntTIozZ5M:&tbnh=123&tbnw=123&prev=/images?q=ferris+wheel&start=20&hl=en&sa=N&as_st=y&ndsp=20&tbs=isch:1

http://www.google.com/imgres?imgurl=http://www.highdisplay.com/wp-content/uploads/2008/06/flat-screen-computer-monitor.JPG&imgrefurl=http://www.highdisplay.com/flat-screen-computer-monitor/&usg=__b1bLwe6PwdsRQJ4FX5qnYjmoPVI=&h=500&w=500&sz=182&hl=en&start=5&itbs=1&tbnid=ta0ghAIf70p5fM:&tbnh=130&tbnw=130&prev=/images?q=computer+screen&hl=en&as_st=y&tbs=isch:1

http://www.google.com/imgres?imgurl=http://www.dregsld.com/tutorials/FloorTile/NewFloorTile1.jpg&imgrefurl=http://www.dregsld.com/tutorials/FloorTile/tiletut01.html&usg=__A8gAqSV9WSkk6lieY2Esd4-bRB4=&h=541&w=600&sz=337&hl=en&start=17&itbs=1&tbnid=IzxpKJVwwq4cyM:&tbnh=122&tbnw=135&prev=/images?q=floor+tile&hl=en&as_st=y&tbs=isch:1

If you play video games…

You may have seen the word POLYGONS, and know it has something to do with graphics.

This is because C.A.D. programs use polygons to render objects C. computer

A. aidedD. drafting

Make a 3-D model


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http://www.google.com/imgres?imgurl=http://www.kamranweb.com/photos/wp-content/uploads/2008/12/halo-3-video-game.jpg&imgrefurl=http://www.kamranweb.com/photos/video-game-wallpapers/halo-3-wallpaper.html&usg=__FqpCf39dlkjuNCsrZR-SUiPYJFI=&h=1200&w=1600&sz=321&hl=en&start=12&itbs=1&tbnid=mjlwuBRci3p0pM:&tbnh=113&tbnw=150&prev=/images?q=video+game&hl=en&as_st=y&tbs=isch:1


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POLYGONS are the basis of most computer imaging.

The more POLYGONS and image has, the higher the quality


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Before we do anything with polygons, you must understand the difference between CONVEX and CONCAVE.

Convex Concave


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It is easier to explain CONCAVE than it is to explain CONVEX

A polygon is CONCAVE if:

There are 2 points somewhere inside the shape

So that if you connect those 2 points with a line

The line goes outside the shape.


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These are all CONCAVE

It may help to think of CONCAVE as having a “cave”or indentation


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If you cannot find 2 points that make a line that goes outside . . .

Then the shape is CONVEX


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And if any of the lines go back into the shape, then it is CONCAVE

Another way to determine if a polygon is CONVEX or CONCAVE is to extend all the sides…


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CONCAVECONCAVE CONCAVE

CONVEX CONVEX CONCAVE

CONVEX CONVEX CONCAVE


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CONVEX or CONCAVE?

These two aren’t even POLYGONS, why?

They have curves.


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In regular geometry, CONCAVE shapes are like your strange cousin.

We just don’t talk about them.

We will spend almost all of our time on CONVEX shapes

CONCAVE IS BAD!!


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What do we call a polygon with three sides?

A triangle


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What do we call a shape with four sides?A quadrilateral


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What do we call a shape with five sides?A pentagon


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What do we call a shape with six sides?A hexagon


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What do we call a shape with seven sides?A heptagon (sometimes also called a septagon)


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What do we call a shape with eight sides?

An octagon


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A nine sided figure is called …

A nonagon

A ten sided figure is called …

A decagon


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While the shapes with more than 10 sides have names, it is acceptable to call them “n-gons”

What does THAT mean?

It means you can call an eleven sided shape an “11-gon”

and a twenty-three sided polygon a “23-gon”


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In a REGULAR polygon…

All the sides are congruent,

All the angles are congruent3

3

3

3

3

1080

1080 1080

1080

1080

EQUILATERAL

EQUIANGULAR


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No matter how many sides the polygon has, they all have the same parts.

SIDEVERTEX

(vertices)


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Naming any polygon has 1 simple rule:Pick any 1 vertex to start,

BC

D

EF

G

A

Then go around the shape, clockwise OR counterclockwise

heptagon BCDEFGAheptagon FEDCBAGTo keep it simple we will try to go as close to alphabetical as we can

heptagon ABCDEFG


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PERIMETER is the distance around the outside of a 2-D object

In other words:If you walked around a polygon, how far would you walk?


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What is the perimeter of this rectangle?

4

12

12

4

Perimeter: 12 + 4 + 12 + 4OR 2(12 + 4)

= 32


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Find the perimeter for each of the following polygons:

4

7

4

7

3

2

3 44 4

44

444

44

31

3 3

3

1

111

1

11

75

2

2

4

2

11

11

11

11

11

228 40

20

22

55

1.

2.

3.

4.

5.

6.


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Find the perimeter for the following REGULAR polygons:

2

20

40 24

60

7.

8.

9.

8

12-gon


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10. Find the perimeter of a regular nonagon with side lengths of 13?

11. Find the perimeter of a regular 27-gon with side lengths of 6?

117

162

POLYGONS and AREAAngles in Polygons

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Interior and

Exterior angles in

Polygons


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The INTERIOR ANGLES of a polygon are the angles inside the figure

POLYGON ANGLES

We know the angles of a triangle add to 180.

In the other shapes, we draw in triangles to find the angle sum.

180

180

180 =360 18

018

0

180

=540

180 18

0

180

180

=720

180

180

180

180

180

180

=1080


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INTERIOR ANGLESTHEOREM: The Sum of the INTERIOR

angles of a convex polygon is (n-2) x 180.(n is the number of sides)

So in a pentagon (5 sides), n=5

The sum of the interior angles:(5-2) x 180

3 x 180

540


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INTERIOR ANGLE

PROBLEMS


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1. In a quadrilateral, what is the sum of the interior angles?

2. In a hexagon, what is the sum of the interior angles?

3. In a decagon, what is the sum of the interior angles?

180)24( 360

180)26( 720

180)210( 440,1


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4.70

90150

x

)1509070(360

50

5.

100

z

110

120105

)100105120110(540 105


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6.

7.

All the angles in the given shape are equal. Find X & Y

All the angles in the given shape are equal. Find X & Y

180180)23(

603180

12060180

Sum of the interior angles

Each of the interior angles

Each exterior angle60120

1080180)28(

13581080

45135180

Sum of the interior angles

Each of the interior angles

Each exterior angle13545


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8. In a regular, convex 12-gon, what is the measure of each interior angle?

9. In a regular, convex 12-gon, what is the measure of each exterior angle?

(12 2) 180 1800 1800

15012

Each interior angle is 150…

So each exterior angle is 30.

What is an EXTERIOR angle?

EXTERIOR ANGLESTHEOREM: The Sum of the EXTERIOR

angles of a convex polygon is 3600

That’s what you get when you extend all the sides in the same direction.


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What is the sum of the exterior angles of an octagon? 360What is the sum of the exterior angles of a pentagon? 360What is the sum of the exterior angles of a decagon? 360

EXTERIOR ANGLESTHEOREM: The Sum of the EXTERIOR

angles of a convex polygon is 3600


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90

68

80

X

63

110

117

X

90 + 68 + 80 + X + 63 = 360 127 + 119 + X = 360

301 + X = 360

X = 59

246 + X = 360

X = 133

13. 14.


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15. What is the measure of an exterior angle of a REGULAR octogon?

16. What is the measure of an exterior angle of a REGULAR decagon?

17. What is the measure of an exterior angle of a REGULAR 36-gon?

18. What is the measure of an exterior angle of a REGULAR 100-gon?

03608

045

036010

036

036036

010

0360100

03.6


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2x

2x+5

x+152x+10

3x-10

3x-10+2x+2x+5+x+15+2x+10 = 360

10x+20 = 360

10x = 340

x = 34

19. FIND X:


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X

20. The picture shows a REGULAR HEXAGON. Find X: 0360

6060


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21. A regular polygon with an unknown number of sides has exterior angles measuring 200. How many sides does it have?

36020

XX X

360 20x20 20

18 x

POLYGONS and AREAArea of Squares and Rectangles

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AREA of

RECTANGLES


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Area is a measure of “flat space”.

If you wanted to cover a floor with 1ft by 1ft tiles, the area of the floor is the number of tiles it takes to cover the floor.


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It is 10 feet long

Here is a small room:

and 6 feet wide

10

6

To cover the room with 1x1 tiles . . .

10 acrossby 6 deep

1

1


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10

6

You need 10 of them across

And 6 deep


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For each of the 10 across, there are 6 deep

10

6

How many total tiles?6 x 10 = 60

This is where base x height comes from

ft

ft


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10

6

Because area is measured by how many “squares” fit in a polygon…

We call the units in area “SQUARE UNITS”

Area is “60 square feet”.

260Area f t


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Formula for the area of a rectangle:

B

HArea Base Height

*Base and height always make a right angle.


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Find the AREA and PERIMETER of each of the following:

1. 2.

70in

35in

14m

70 35Area 22,450in

70 35 70 35Perimeter 210in

14 14Area 2196m

14 14 14 14Perimeter 56m


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3. Find the area of the shape shown.

12

9

3

3

To do this, find the area of the big blue piece (pretend it doesn’t have a hole in it)

Then find the area of the “cutout”

Finally, subtract them.

9 12A 108 2u

3 3A 9 2u

108 9 299u


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4. Find the AREA and PERIMETER of the shape shown.

6cm

2cm1cm

2cm

1cm

12cm

6cm

Perimeter is easy, just add up the sides.

The only trick is that you have to make sure you have all the sides.

6 2 1 2 6 8 1 12P

8cm 38cm


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6cm

2cm1cm

2cm

1cm

12cm

6cm

8cm

To find the area, cut the shape into parts you can work with

22cm

224cm

28cm

2 24 8A 234cm


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6cm

2cm1cm

2cm

1cm

12cm

6cm

8cm

To find the area, cut the shape into parts you can work with

210cm

212cm

212cm

12 12 10A 234cm

POLYGONS and AREAArea of Triangles

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AREA of TRIANGLES


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Find the area of the rectangle shown:

4m

10

m

4m

10

m

Area = BxH

=4x10

=40m2

4m

10

m

4m

10

m

Each of these triangles is HALF the area of the original rectangle.

20m2

20m2


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What if it is not a right triangle?


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No matter the type of triangle… …It is still HALF of a RECTANGLE.

b

h


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Formula for finding the area of a triangle:

12

A b h

*Base and height always make a right angle.

b

h

b b

hh


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Find the AREA and PERIMETER for each of the following triangles:

5m3m

4m

EXAMPLE #1

To find the area, we need base and height

hbArea 2

1

Remember the base and height make a right angle with each other.

342

1Area 26m

For perimeter, just add all the sides:

543 Perimeterm12


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15ft

8ft17ft

EXAMPLE #2

hbArea 2

1

8152

1Area 260 ft

For perimeter, we need to know that 3rd side.

We can find it using the PYTHAGOREAN THEOREM

2 2 28 15 c 17 c

8 15 17Perimeter ft40


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10 10

12

8EXAMPLE #3

hbArea 2

1

1282

1Area 248u 101012 Perimeter

u32


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STUDENT PROBLEMS

1#

Find the AREA and PERIMETER for each of the following triangles 2#

12m

16m20m

24ft

10ft

161221: A

296m

201612: Pm48

26

102421: A

2120 ft

262410: P60f t


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Find the AREA for this triangle

5

8

712

EXAMPLE #4This is an easy one…

…Sometimes the height is outside the triangle…

…but that doesn’t change anything.

hbArea 2

1

582

1Area 220u

12 8 7Perimeter 27u


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Find the AREA for this triangle

26 26

48

EXAMPLE #5

This triangle is isosceles.

That means the height to the base bisects the base.

24 24

We need to find the height.

h

We will have to use the PYTHAGOREAN THEOREM to find the height. 222 2624 h

10h

10

hbArea 2

1

10482

1Area 2240u


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STUDENT PROBLEMS

Find the AREA and PERIMETER for each of the following triangles3#

16in

17in17in

151621: A

2120in

171716: Pin50

15

8

4#

6

594

3

6421: A

212u

965: Pu20


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Find the area of this shape:

12in

9in 14in

12in

13in5in

30in2

108in2

138in2

EXAMPLE #6


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STUDENT PROBLEMS

Find the AREA of the following shape.5#

17in

15in

9in

17in 15

8

135

60

2135 60 195in


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These 2 triangles are similar, with a scale factor of

If the area of the big one is 50cm2 then what is the area of the small one?

35

20cm

5cm

12cm

3cm

13 12

2A

18

EX

AM

PLE

#7


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35

X

Y

X5

3

Y5

3

XYArea2

1100 Area

5

3 YX

5

3

2

1

EX

AM

PLE

#7


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35

X

Y

X5

3

Y5

3

XYArea2

1100 Area

5

3 YX

5

3

2

1Area

5

3 YX

5

3

2

1

2100sArea

3 3100

5 5Area

36Area


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Finding the AREA of SIMILAR POLYGONS

If 2 polygons are similar, the ratio of their areas is the square of the scale factor

Area = 36m2

These 2 polygons are similar.The scale factor is

3

2

If the area of the big one is 36, find the area of the other.

3

2

3

236 Area

216m


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STUDENT PROBLEMS

ABCDEofareathefindUWXYZABCDE ,~

E

A

B

CD

Z

U

W

XY

A: 54m2

68

First, find the scale factor4

3

8

6

Multiply the area by the scale factor twice 3

4

3

454 296m

6#


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Find the area of this regular pentagon

8

12

10

6

24

EXAMPLE #8


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Find the area of this regular pentagon

8

12

10

6

242424

24

24

24

24 24

24

24

24

Area = 24x10

=240u2

EXAMPLE #8


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STUDENT PROBLEMS

Find the AREA of this regular octogon:

10m

12m

5

Area of 1 triangle:

# of triangles:

30

16

Area: 30x16=480m2

7#

POLYGONS and AREAArea of Parallelograms

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Area of PARALLELOGRAMS


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To calculate the area of a parallelogram…

Just Multiply base and height

B

H

Area of a Parallelogram

hbArea h

b*Base and height make a right angle.


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Area of a

RHOMBUS


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d1

d2

Calculating the area of a rhombus can be done the same as a parallelogram…OR you can use the

diagonals


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d1

d2

Calculating the area of a rhombus can be done the same as a parallelogram…OR you can use the

diagonals

This rectangle has an area of

A = d1 x d2

So each Rhombus is half that.


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•It does NOT matter which diagonal is which.

•Remember the diagonal goes all the way across the shape.

•You will frequently be given only half of a diagonal.

Area of a Rhombus

212

1ddArea

*the diagonals always make a right angle.

d1

d 2


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Find the area of each Parallelogram

#1 #2

10ft

20ft

8ft

16cm

7cm

22cm

hbA 208 2160 ft

hbA 167 2112cm


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Find the area of each Rhombus

#3 #4 #5

3m4m

127

10ft

20ft 12

7

212

1ddA

862

1A

224mA

20102

1A

2100 ftA

14242

1A

2168A u

POLYGONS and AREAArea of Trapezoids

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Area of a

Trapezoid


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This is a trapezoid: It has 1 set of parallel sides.

The MIDSEGMENT …

Joins the midpoints of the legs

Base1

Base2Le

g Leg

Has a length that is the average of the bases

211 bb

midsegment


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Like most shapes, the area of a trapezoid is based on a rectangle

midsegment:

heig

ht:

Base1

Base2


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Area of a Trapezoid

heightmidsegmentArea

hbb

2

21

1b

2b

h


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Find the area of each of the following trapezoids:

#1 #2 #3

8m

6m

7m 12mi

11mi

6mi

13mi

9

15

8

10

72

68

A

249m

122

611

A

2102mi

82

159

A

296u


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Find the area of this shape:

22

10

11

2022

9

242

144144 + 242 =

386u2


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Find the area:


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Find the area:


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Find the area:


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Find the area:


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Find the area:


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Find the area:


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Find X:


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Find X:


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Find X:


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Find X:


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Find X:


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Find X:


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Find X:


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Find X:


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Find X:

POLYGONS and AREACircumference and Area of Circles

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RadiusDiameterChordSecantTangentPoint of Tangency

Distance from center to edge of a circleDistance from edge to edge of a circle through the

centerAny line segment that goes from edge to edge in a

circleAny line that passes through a circle

A line that touches the circle at exactly 1 point

The point where a circle an tangent touch


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C

A

BD

E

F

G

H

J


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C

A

BD

E

F

G

H

AC

J


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C

A

BD

E

F

G

H

BD

J


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C

A

BD

E

F

G

H

ED

J


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C

A

BD

E

F

G

H

GH

J


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C

A

BD

E

F

G

H

JF

J


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C

A

BD

E

F

G

H

F

J


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For a circle, the formulas for area and perimeter are different, because there are no sides and there is no base or height.


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5

Radius:

Diameter:

Circumference:

The distance from the center of a circle to the edge

The distance from edge to edge of a circle, passing through the center

10

The distance around the outside of a circle


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10

What is pi Pi is what you get if you divide the distance around the outside of any circle by that circles diameter


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Area of a circle:

Circumference of a circle:

2rA

rC 2

5

25A

25

52C10


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8

Find the area and circumference:

AREA CIRCUMFERENCE2rA 28A

64A

rC 2

82C

16C


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12

Find the area and circumference:

AREA CIRCUMFERENCE2rA 26A

36A

rC 2

62C

12C

6

Notice that this is just the diameter times pi.

DrC 2


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FINDING ARCLENGTH

Here is a slice of pizza.

60

6It has a 6 inch radius,And we cut a 60 degreeSlice out of it.

How much crust do you have ?

The crust of the whole pizza:(circumference)

rC 2 62 12

68.37

But you don’t have the wholePizza, you just have 60 degrees

360

6068.37 inches28.6


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To find ARCLENGTH

360

2angle

rarclength120

8

360120

82

7.16


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Find the arclength.

70

5

360

2angle

rarclength

360

7052

1.6


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Find the area of the SECTOR


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2rArea of the whole circle:

Area of a sector or “slice”:

2

360

angler 2

360r

2 606

360 218.84in

Find the area of the SECTOR


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2

360r

2 9010

360

278.5f t

POLYGONS and AREAOPENERS

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ABCDEFGHIJKLMNOPQRS

TUVWXYZ

POLYGONS and AREAOPENERS A

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POLYGONS and AREAOPENERS B

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POLYGONS and AREAOPENERS C

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POLYGONS and AREAOPENERS D

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POLYGONS and AREAOPENERS E

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POLYGONS and AREAOPENERS F

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POLYGONS and AREAOPENERS G

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Can you do this WITHOUT

a CALCULATOR?

POLYGONS and AREAOPENERS H

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In your job as a cashier, a customer gives you a $20 bill to pay for a can of coffee that costs $3.84. How much change should you give back?

a) $15.26 b) $16.16 c) $16.26 d) $16.84 e) $17.16

POLYGONS and AREAOPENERS I

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How much time is there between 7:35 a.m. and 5:25 p.m.?

a) 8 hours and 50 minutesb) 9 hours and 10 minutesc) 9 hours and 50 minutesd) 10 hourse) 10 hours and 50 minutes

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POLYGONS and AREAReviews

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Find the Area (assorted)Compound Polygon (5-Questions)Jeopardy ReviewReview for Quiz 8.1-8.2Review for Quiz 8.3-8.5Review Harder Problems

Geoemtry NAME ______________________ Find the area of each composite shape shown. All angles are right. 1. 2. 3. 4.

5.


5.


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Find the area of each shape:1 2 3 4

5 6 7 8

9 10 11

24 6 12154

69 28 31 126

12

29 52

3

answers

500500500500500

400 400400400400400

500

300300300300300300

200200200200200200

100100100100100100

Column 6Column 5Column 4Column 3Column 2Column 1


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5.


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5.


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Review for quiz 8.1-8.2

VOCABULARY: Equilateral, Equiangular, Regular, Triangle, Quadrilateral, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon, Convex, Concave, Interior, Exterior


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Identify as CONVEX or CONCAVE


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What is the sum of the interior angles of a decagon?

What is the sum of the exterior angles of a pentagon?

x4


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Find X

x4

90

68

80

X

63

90 + 68 + 80 + X + 63 = 360

301 + X = 360

X = 59

130

110 112

138

X113

X+113+130+110+112+138=720

X=117


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What is the measure of each interior angle of a regular 30-gon?

What is the measure of each exterior angle of a regular hexagon?

x4

A regular polygon has exterior angles measuring 150. How many sides does it have?

POLYGONS and AREAReview 8.3-8.5

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POLYGONS and AREAReview HARDER PROBLEMS

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#1 Find the area of the RED region


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#2 Find the measure of X.


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#3 Find the area:


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#4


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#5 Find the measure of the area of the sector.


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#6 The area of the square shown is 144 in2. Find the length of the sides.


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#7 Find X if the area of the trapezoid is 48in2:

X

4in

12in


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#8 Find the area of the RED region:

Documents

POLYGONS and AREA Classifying Polygons Angles in Polygons Area of Squares and Rectangles Area of Triangles Area of Parallelograms Area of Trapezoids Circumference