Chapter 11 Areas 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.

Math Notation for Different Measurements

Dimensions

• Length (1 dimension) – The length of a line is….

Notation

• 1 unit - 2cm - 3in • Area (2 dimensions) – The area of a rectangle is ….

• 2 units 2 • 3 cm 2 – 10 in 2 • Volume (3 dimensions) – The volume of a cube is….

• 4 units 3 • 8 cm 3

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

What types of jobs use area everyday?

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

s s Area = s 2 • •

What’s the are of a square with..

side length of 4?

perimeter of 12 ?

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

A B

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

With you partner: Why would congruent figures have the same area?

Area Addition Postulate The area of a region is the sum of the areas of its non-overlapping parts. Area of figure = Area A + Area B + Area C B A C

Base (

b

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

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

Called Height

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

Area = b x h

Using the variables shown on the diagram create an equation that would represent the perimeter of the figure.

b h

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 h b b h A 12m 3m 9cm 54 cm 2 y-2 y

b b h A Group Practice h 12m 3m 36m 2 9cm y-2 6cm y 54 cm 2 y 2 – 2y

Find the area of the rectangle 3 5

AREA = 12

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

3 2 4 5 5

A = 114 units

2 6

11-2: Areas of Parallelograms, Triangles, and Rhombuses Objectives • Determine and apply the area formula for a parallelogram, triangle and rhombus.

Base (b) and Height (h)

PARTNERS….

• How do a rectangle and parallelogram relate?

• What could I do with this parallelogram to make it look like a rectangle?

h b

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

**This right triangle is key to helping solve!!

h

Area = b x h

b

Triangle Demo • How can I take two congruent triangles and connect them to make a new shape?

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

A = bh 2

h b

Partners • How would you label the base and height of these triangles?

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

d 1

d 1

∙

d 2

2

d 2 **WHAT DO YOU SEE WITHIN THE DIAGRAM?

White Board Practice • Find the area of the figure

A

 9 6 3 3 60º 3 6

White Board Practice

A

 12 5 5 6 • Just talk about this one

5 White Board Practice • Find the area of the figure 12

A

 30 13 • • Just talk about this one Just talk about this one

White Board Practice • Find the area of the figure

A

 20 2 5 2 5 • Just talk about this one

White Board Practice • Find the area of the figure

A

 4 4 4 3 4

White Board Practice • Find the area of the figure 5

A

 24 4 5 5 4 5

Organization is Key

• Always draw the diagrams • Know what parts of the formula you have and what parts you need to find • Right triangles will help you find missing information

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 median leg base leg What type of trap do we have if the legs are congruent?

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

b 2 median • Length of median = ½ (b 1 +b 2 ) b 1 The length of the median is the_______ of the bases.

Height • The height of the trapezoid is the segment that is perpendicular to the bases of the trapezoid b 2 How do we measure height for a trap?

h

Partners: Why is the height perpendicular to both bases?

b 1

Labeling Height for Isosceles Trap • Always label 2 heights when dealing with an isosceles trap

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

b 2 h b 1 demo

White Board Practice 1. Find the area of the trapezoid and the length of the median 7 5 A = 50 Median = 10 13

White Board Practice 3. Find the area of the trapezoid and the length of the median 14 A = 138 Median = 11.5

12 13 9

Group Practice • Find the area of the trapezoid 8 8 8 60º

Area =

48 3

Group Practice • Find the area of the trapezoid 45º 3 2 4

Area =

33 2

Group Practice • Find the area of the trapezoid 12 30º 30º 30

Area =

63 3

11.4 Areas of Regular Polygons Objectives • Determine the area of a regular polygon.

Regular

Polygon Review •All sides congruent •All angles congruent

(n-2) 180 n

side

Circles and Regular Polygons • Read Pg. 440 and 441 – Start at 2 nd paragraph, “Given any circle… • What does it mean that we can inscribe a poly in a circle?

–

Each vertex of the poly will be on the circle

Center

of a regular polygon is the center of the circumscribed circle center

Radius

of a regular polygon is the distance from the center to a vertex is the radius of the circumscribed circle

Central angle

of a regular polygon Is an angle formed by two radii drawn to consecutive vertices Central angle How many central angles does this regular pentagon have?

How many central angles does a regular octagon have?

Think – Pair – Share

Central angle What connection do you see between the 360◦ of a circle and the measure of the central angle of the regular pentagon?

360 n

Apothem

of a regular polygon the perpendicular distance from the center to a side of the polygon apothem How many apothems does this regular pentagon have?

How many apothems does a regular triangle have?

Regular

Polygon Review

**What do you think the apothem does to the central angle?

central angle center apothem

Perimeter = sum of sides

side

Theorem The area of a

regular

polygon is half the product of the apothem and the perimeter.

What does each letter represent in the diagram?

r s a

s = length of side p = 8s A = ap 2

RAPA • R adius • A pothem • P erimeter • A rea r a s This right triangle is the key to finding each of these parts.

Radius, Apothem, Perimeter 1. Find the central angle 360 n

Radius, Apothem, Perimeter 2. Draw in the apothem… This divides the isosceles triangle • into two congruent right triangles

How do we know it’s an isosceles triangle?

Radius, Apothem, Perimeter 3. Find the missing pieces •

What does ‘x’ represent?

r a x

Radius, Apothem, Perimeter • Think 30-60-90 • Think 45-45-90 • Think SOHCAHTOA

A = ½ ap r 8 a p A 1. Central angle 2. ½ of central angle 3. 45-45-90 30-60-90 SOHCAHTOA r x a

A = ½ ap r x a r 8 a p A 4 24 3 48 3 1. Central angle 2. ½ of central angle 3. 45-45-90 30-60-90 SOHCAHTOA IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE?

A = ½ ap r 5 2 a p A 1. Central angle 2. ½ of central angle 3. 45-45-90 30-60-90 SOHCAHTOA r a x

A = ½ ap r a 5 2 5 p A 40 100 r a x IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE?

A = ½ ap r 8 a p A 1. Central angle 2. ½ of central angle 3. 45-45-90 30-60-90 SOHCAHTOA r a x

A = ½ ap r a p A 8 4 3 48 96 3 1. Central angle 2. ½ of central angle 3. 45-45-90 30-60-90 SOHCAHTOA r a x

A = ½ ap r a p 24 3 A 1. Central angle 2. ½ of central angle 3. 45-45-90 30-60-90 SOHCAHTOA r a x

A = ½ ap r a p A 4 3 6 24 3 72 3 1. Central angle 2. ½ of central angle 3. 45-45-90 30-60-90 SOHCAHTOA r a x

11.5 Circumference and Areas of Circles Objectives • Determine the circumference and area of a circle.

 

C d

 3.1415

r

 • Greek Letter Pi (pronounced “pie”) – Used in the 2 main circle formulas: • Circumference and Area

(What are these?)

• Pi is the ratio of the circumference of a circle to the diameter.

• Ratio is constant for

ALL CIRCLES

• Irrational number (cannot be expressed as a ratio of two integers) • Common approximations – 3.14

– 22/7

Circumference

The distance around the outside of a circle.

**The

Circumference

and the

diameter

have a special relationship that lead us to 

=

C d

Circle Number Experiment Circumference (nearest mm) Diameter (nearest mm) Ratio of Circumference/Diameter (as a decimal) 1 2 3 4 5

Group Experiment

1. With the circular object 2. 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 diameter 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 hundredth.

8. Pass you circular object to the next group and repeat

Circle Number Experiment 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?

Circumference

The distance around the outside of a circle.

C = ∏ d C = ∏ 2r

r r d C = circumference r = radius d = diameter

Area

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

A

 

r

2 B r For both formulas always leave answers in 

r

15 WHITEBOARDS *put answers in terms of pi

d C

8 26∏

A

100∏ 18∏

Quiz review - Set up these diagrams 1. A square with side 2√3 2. A rectangle with base √4 and diagonal √5 3. A parallelogram with sides 6 and 10 and a 45◦ angle 4. A rhombus with side 10 and a diagonal 12 5. An isosceles trapezoid with bases of 2 and 6 and base angles that measure 45 ◦ 6. A regular hexagon with a perimeter 72

11.6 Arc Length and Areas of Sectors Objectives • Solve problems about

arc length

and

sector and segment area

.

A r B

Warm - up 1. If you had the two pizzas on the right and you were really hungry, which one would you take a slice from? Why?

Same angle

Arc Measure

the measure of an arc is given by the measure of its central angle.

A 80 C The central angle tells us how much of the 360 ◦ of crust we are using from our pizza.

AC mAC



80

 B The central angle measure and arc measure are the same no matter the size of circle.

C Remember Circumference The distance around the outside of a circle.

x ◦ x ◦ r B Finding the total

length

C

 

d

 2 

r

Arc Length

The length of the arc is part of the circle’s circumference… the question is, what fraction of the total circumference does it represent? x ◦ Circumference of circle Degree measure of arc x ◦ O LENGTH OF ARC 

x

( 360 )  2 

r

Example If r = 5, what is the length of CB? Measure of CB = 60 ◦ 60 = 1 360 6 C 1 6 (2 

∙ 5) =

5 3  60 ◦ O B

A

 

r

2 Remember Area C

Sector of a circle aka – the area of the piece of pizza

B

Area of a Sector

The area of a sector is part of the circle’s area… the question is, what fraction of the total area does it represent? x ◦ Area of circle Degree measure of arc x ◦ O AREA OF SECTOR  (

x

360 )  

r

2

Example If r = 5, what is the area of sector COB? Measure of CB = 60 ◦ 60 = 1 360 6 C B 60 ◦ O 1 6 ( 

∙ 5 2 ) =

6

REMEMBER!!!

• Both arc length and the area of the sector are different with different size circles! • Just think pizza

WHITEBOARDS • ONE PARTNER OPEN BOOK TO PG. 453 (classroom exercises) • ANSWER #2 – Length = 4  – Area = 12  • ANSWER # 4 – Length = 6  – Area = 12  • ANSWER #1 (we)

WHITEBOARDS • Find the area of the shaded region B • 25 ∏ - 50 10 A 10 O

11-7 Ratios of Areas Objectives • Solve problems about the ratios of areas of geometric figures.

Ratio • A

ratio

of one number to another is the quotient when the first number is divided by the second.

• A comparison between numbers • There are 3 different ways to express a ratio

1 2 1 : 2 1 to 2 3 5 3 : 5 3 to 5 a b a : b a to b

Solving a Proportion 3 5 

a

15 5

a

 45

a

 9 First, cross-multiply Next, divide by 5

The Scale Factor • • If two polygons are similar, then they have a

scale factor The reduced ratio between any pair of corresponding sides or the perimeters.

• 12:3 

scale factor of 4:1

12

**What have we used scale factor for in past chapters?

3

Comparing Areas of Triangles

Two triangles with equal heights 4 4

Two triangles with equal heights Partners: Compare the ratio of the areas to the ratio of the bases

A

 1 2

bh

4 4 7 3

7 4 Ratio of their areas 14 4 6 3

7 4 Ratio of areas = ?

7 4 3 3

Comparing Areas of Triangles

Rule # 1

If two triangles have

equal heights….

4 4

then the ratio of their

areas

equals the ratio of their

bases.

Two triangles with equal bases Partners: Compare the ratio of the areas to the ratio of the heights 8 2 5 5

5 8 Ratio of Areas 20 5 2 5

5 8 Ratio of Areas = ?

4 1 2 5

Comparing Areas of Triangles

Rule # 2

If two triangles have

equal bases….

5 5

then the ratio of their

areas

equals the ratio of their

heights.

Comparing Areas of Triangles

Rule # 3

If two triangles are

similar….

6 X

~

Y 4

With your partner:

What is the ratio of the areas of X to Y?

then the ratio of their

areas

equals the square (

2

) of the

scale factor.

Theorem

If the

scale factor

of two similar figures is a:b, then…

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

~

Area = 27

Scale Factor Ratio of P Ratio of A

7: 3 – 7: 3 – 49 :9 7 3

WHITEBOARDS • • •

OPEN BOOK TO PG. 458

(classroom exercises) – ANSWER #4 Ratio of P – 1:3 – – Ratio of A – 1:9 If the smaller figure has an area of 3 what is the area of the larger shape?

– ANSWER # 10 Scale factor – 4:7 – Ratio of P – 4:7 a.

ANSWER # 13 No b. ADE ~ ABC c. 4: 25 d. 4:21

WHITEBOARDS • The areas of two similar triangles are 36 and 81. The perimeter of the smaller triangle is 12. Find the perimeter of the bigger triangle. • 36/81 = 4/9  • 2/3 = 12/x  2/3 is the scale factor x = 18

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

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

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 F

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

Remember • Scale Factor a:b • Ratio of perimeters a:b • Ratio of areas a 2 :b 2

11-8: Geometric Probability Solve problems about geometric probability

Read Pg. 461

• Solving Geometric Problems using 2 principles 1. Probability of a point landing on a certain part of a line (length) 2. Probability of a point landing in a specific region of an area (area)

Sample Space

The number of all possible outcomes in a

1. Total length of the line 2. Total area

random experiment.

Event

:

A possible outcome in a random experiment.

1. Specific segment of the line 2. Specific region of an area

Probability

The calculation of the possible outcomes in a random experiment



For example: When I pull a popsicle stick from the cup, what is the chance I pull your name?

Geometric Probability

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

•

In a 10 minute cycle a bus pulls up to a hotel and waits for 2 minutes while passengers get on and off. Then the bus leaves. If a person walks out of the hotel front door at a random time, what is the probability that the bus is there?

Geometric Probability

2. The area of an event divided by the area of the sample space.

• If a beginner shoots an arrow and hits the target, what is the probability that the arrow hits the red bull’s eye?

1 2 3

WHITEBOARDS • •

OPEN BOOK TO PG. 462

– ANSWER #2 1 / 3 – ANSWER # 3 Give answer in terms of pi (classroom exercises)

WHITEBOARDS • Find the ratio of the areas of WYV to XYZ –

4 to 49

• Find the ratio of the areas of WYV to quad WVZX –

4 to 45

• Find the probability of a point from the interior of XYZ will lie in the interior of quad XWYZ –

45/49

5 W 2 Y X V Z

Drawing Quiz- Set up these diagrams 1. A rectangle with base 10 and diagonal 15 2. A parallelogram with sides 6 and 10 and a 60◦ angle 3. A rhombus with side 10 and a diagonal 12 4. An equilateral triangle with a perimeter = 27 5. Sector AOB: AO = 12 and the central angle equals 50 degrees 6. Isosceles triangle with base of 10 and perimeter of 40.

Chapter Review • 16 • 12 • 21 • 22 Test Review • Chapter test – 4 – 9 – 12 – 15