Chapter 11 Areas of Plane Figures

Chapter 11Areas of Plane Figures

• Understand what is meant by the area of a polygon.

• Know and use the formulas for the areas of plane figures.

• Work geometric probability problems.

11-1: Area of Rectangles

Objectives

• Learn and apply the area formula for a square and a rectangle.

Area

A measurement of the region covered by a geometric figure and its interior.

Theorem

The area of a rectangle is the product of the base and height.

b

hArea = b x h

Base

• Any side of a rectangle or other parallelogram can be considered to be a base.

Altitude

• Altitude to a base is any segment perpendicular to the line containing the base from any point on the opposite side.

• Called Height

Postulate

The area of a square is the length of the side squared.

s

s

Area = s2

If two figures are congruent, then they have the same area.

Postulate

A B

If triangle A is congruent to triangle B, then area A = area B.

Find the area

Area Addition Postulate

The area of a region is the sum of the areas of its non-overlapping parts

A

B

C

Remote Time

Classify each statement as True or False

Question 1

• If two figures have the same areas, then they must be congruent.

Question 2

• If two figures have the same perimeter, then they must have the same area.

Question 3

• If two figures are congruent, then they must have the same area.

Question 4

• Every square is a rectangle.

Question 5

• Every rectangle is a square.

Question 6

• The base of a rectangle can be any side of the rectangle.

White Board Practice

b 12m 9cm y-2

h 3m y

A 54 cm2

b

h


b 12m 9cm y-2

h 3m 6cm y

A 36m2 54 cm2 y2 – 2y

b

h

Group Practice

• Find the area of the figure. Consecutive sides are perpendicular.

54

23

6

5

Group Practice

• Find the area of the figure. Consecutive sides are perpendicular.

54

23

6

5

A = 114

11-2: Areas of Parallelograms, Triangles, and Rhombuses

Objectives

• Determine and apply the area formula for a parallelogram, triangle and rhombus.

Tons of formulas to memorize

• So don’t memorize them

• Understand them !

Refresh my memory…

What is the area of a rectangle ?

Refresh my memory…

what is the height in a rectangle?

Altitude to a base is any segment perpendicular to the line containing the base from any point on the opposite side.

h

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h

h

h

Cut Slide over and tape

h

Do we have any leftover paper?

So this means the area must be the same

Theorem

The area of a parallelogram is the product of the base times the height to that base.

b

h

Area = b x h

But Wait….

What do we have ?

X 2

Theorem

The area of a triangle equals half the product of the base times the height to that base.

b

h

1

2Area b h

Theorem

The area of a rhombus equals half the product of the diagonals.

d1

d2

1 2

1

2Area d d

Remote Time


• Find the area of the figure

4

4

4



4

4

4

34A



3

6

6

3

60º



3

6

6 39A

3

60º



5 5

6



5 5

12A

6



2

25

5



20A2

25

5



4

5

4

55

5



24A4

5

4

55

5



12

13

5



30A12

13

5

11-3: Areas of Trapezoids

Objectives

• Define and apply the area formula for a trapezoid.

Trapezoid Review

A quadrilateral with exactly one pair of parallel sides.

base

base

leg leg

height

median

Median

• Remember the median is the segment that connects the midpoints of the legs of a trapezoid.

• Length of median

= ½ (b1+b2)b1

b2

median

Height

• The height of the trapezoid is the segment that is perpendicular to the bases of the trapezoid

b1

h

b2

Theorem

The area of a trapezoid equals half the product of the height and the sum of the bases.

b1

h

b2 1 2

1( )

2Area h b b

Sketch


13

5

71. Find the area of the trapezoid and the length of the median


13

5


A = 50Median = 10


13

6


10


13

6


10

A = 54Median = 9

White Board Practice3. Find the area of the trapezoid and the length of the median

14

12

13

9

White Board Practice3. Find the area of the trapezoid and the length of the median

14

12

13

9

A = 138Median = 11.5

Group Practice

• A trapezoid has an area of 75 cm2 and a height of 5 cm. How long is the median?

Group Practice

• A trapezoid has an area of 75 cm2 and a height of 5 cm. How long is the median?

Median = 5 cm

Group Practice

• Find the area of the trapezoid

8 8

8

60º

Group Practice


8 8

8

60º

Area = 348

Group Practice


45º

23

4

Group Practice


Area = 2

33

45º

23

4

Group Practice


12

30

30º 30º

Group Practice


Area = 363

12

30

30º 30º

11.4 Areas of Regular Polygons

Objectives

• Determine the area of a regular polygon.

Regular Polygon Review

side

All sides congruentAll angles congruent

Center of a regular polygon

center

is the center of the circumscribed circle

Radius of a regular polygon

center

is the radius of the circumscribed circle

is the distance fromthe center to a vertex

Central angle of a regular polygon

Central angle

Is an angle formed bytwo radii drawn to consecutive vertices

Apothem of a regular polygon

apothem

the perpendicular distance from the center to a side of the polygon

Regular Polygon Review

side

center

radiusapothem

central angle

Perimeter = sum of sides

Polygon Review ContinuedSides Name 1 interior 1 Central 3 Triangle 60 120

4 Square 90 90

5 Pentagon 108 72

6 Hexagon 120 60

7 Septagon 128.6 51.4

8 Octagon 135 45

n n-gon (n-2)180 360

n n

TheoremThe area of a regular polygon is half the product of

the apothem and the perimeter.

a

s

r

P = 8s

1

2Area ap

RAPA

• R adius

• A pothem

• P erimeter

• A rea

a

s

r

Radius, Apothem, Perimeter

1. Find the central angle 360 n


2. Divide the isosceles triangle into two congruent right triangles


ra

3. Find the missing piecesx


• Think 30-60-90

• Think 45-45-90

• Thing SOHCAHTOA

r a p A

ra

x

1. Central angle2. ½ of central angle3. 45-45-90

30-60-90SOHCAHTOA

25A = ½ ap

r a p A

5 40 100

ra

x

25A = ½ ap

r a p A

ra

x


30-60-90SOHCAHTOA

3A = ½ ap

r a p A

12

ra

x


30-60-90SOHCAHTOA

3A = ½ ap

6 38

r a p A

8

ra

x


30-60-90SOHCAHTOA

A = ½ ap

r a p A

8 4


30-60-90SOHCAHTOA

A = ½ ap324

ra

x

348

r a p A


30-60-90SOHCAHTOA

A = ½ ap36

ra

x

r a p A

2 1


30-60-90SOHCAHTOA

A = ½ ap36

ra

x

33

r a p A

8

r a

x


30-60-90SOHCAHTOA

A = ½ ap

r a p A

8 48


30-60-90SOHCAHTOA

34A = ½ ap

396

r a

x

r a p A


30-60-90SOHCAHTOA

A = ½ ap324

r a

x

r a p A

6


30-60-90SOHCAHTOA

372A = ½ ap

34 324

r a

x

11.5 Circumference and Areas of Circles

Objectives

• Determine the circumference and area of a circle.

3.1415C

d

r

r a p A ??

Doesn’t work! Why?

r a p A ??

Doesn’t work! Why?

CircumferenceThe distance around the outside of a circle.

Experiment

1. Select 5 circular objects2. Using a piece of string measure around

the outside of one of the circles.3. Using a ruler measure the piece of string

to the nearest mm.4. Using a ruler measure the diamter to the

nearest mm.5. Record in the table.

Experiment

6. Make a ratio of the Circumference.

Diameter

7. Give the ratio in decimal form to the nearest hundreth.

ExperimentCircle Number

Circumference (nearest mm)

Diameter (nearest mm)

Ratio of Circumference/Diameter (as a decimal)

1

2

3

4

5

What do you think?

1. How does the measurement of the circumference compare to the measurement of the diameter?

2. Were there any differences in results? If so, what were they?

3. Did you recognize a pattern? Were you able to verify a pattern?

• Greek Letter Pi (pronounced “pie”)• Pi is the ratio of the circumference of a circle to

the diamter.• Ratio is constant for all circles• Irrational number• Common approximations

– 3.14

– 3.14159

– 22/7

CircumferenceThe distance around the outside of a circle.

2C d r r

Area

B

The area of a circle is the product of pi times the square of the radius.

2A rr

11.6 Arc Length and Areas of Sectors

Objectives

• Solve problems about arc length and sector and segment area.

r

A

B

Remember Circumference

C = 2r

Arc LengthThe length of an arc is the product of the

circumference of the circle and the ratio of the circle that the arc represents.

O

B

C

x r

rx

LengthBC 2360

Remember Area ?

2rA

Sector AreaThe area of a sector is the product of the area

of the circle and the ratio of the circle that the sector of the circle represents.

A

B

C x r

2

360r

xorBCAreaofSect


11-7 Ratios of Areas

Objectives

• Solve problems about the ratios of areas of geometric figures.

Comparing Areas of Triangles

Two triangles with equal heights

4 4


bhA2

1

4 4


bhA2

1

4 4

7 3

Ratio of their areas

4 4

7 3

6

14

Ratio of areas = ?

4 4

7 3

3

7

If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.

Two triangles with equal bases

5 5

8

2

Ratio of Areas

5 5

8

2

5

20

Ratio of Areas = ?

5 5

8

2

1

4

If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights.

If two triangles are similar, then the ratio of their areas equals the square of their scale factor.

TheoremIf the scale factor of two similar triangles is a:b, then

1.)the ratio of their perimeters is a:b 2.)the ratio of their areas is a2:b2.


• Find the ratio of the areas of ABC: ADB

A

B

CD


• Find the ratio of the areas of ABD: BCD

A

B

CD

Remember

• Scale Factor a:b

• Ratio of perimeters a:b

• Ratio of areas a2:b2

Remote Time

• True or False

T or F

If two quadrilaterals are similar, then their areas must be in the same ratio as the square of the ratio of their perimeters

T or F

If the ratio of the areas of two equilateral triangles is 1:3, then the ratio of the perimeters is 1: 3

T or F

If the ratio of the perimeters of two rectangles is 4:7, then the ratio of their areas must be 16:49

T or F

If the ratio of the areas of two squares is 3:2, then the ratio of their sides must be 2:3

11-8: Geometric Probability

Solve problems aboutGeometric probability.

Event: A possible outcome in a random experiment.

Sample SpaceThe number of all possible outcomes in a random experiment.

Probability•The calculation of the possible outcomes in a random experiment.

( )

Event SpaceP e

Sample Space

Geometric Probability• The area of the event divided by the area of

the sample space.

• The length of an event divided by the length of the sample space.

Homework Set 11.8

Pg 463

(2-10 even

Pg 465 (1-10)

Documents

Chapter 11 Areas of Plane Figures