Transcript Areas of Plane Figures Areas of Plane Figuresx

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)
Notation
• 1 unit - 2cm - 3in
– The length of a line is….
• Area (2 dimensions)
– The area of a rectangle is ….
• Volume (3 dimensions)
– The volume of a cube is….
• 2 units2
• 3 cm2– 10 in2
• 4 units3
• 8 cm3
Area
A measurement of the region covered by a geometric
figure and its interior.
What types of
jobs use area
everyday?
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
Finding area? Ask these
questions…
1. What is the area formula for this shape?
2. What part of the formula do I already
have?
3. What part do I need to find?
4. How can I use a right triangle to find the
missing part?
Postulate
The area of a square is the length of the side squared.
Area = s2
s
s
What’s the are of a square with..
• side length of 4?
• perimeter of 12 ?
Theorem
The area of a rectangle is the product of the base and
height.
h
Area = b x h
Using the variables shown on
the diagram create an equation
that would represent the
perimeter of the figure.
b
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
12m
h
3m
A
P
9cm
y-2
y
54 cm2
Group Practice
h
b
b
12m
9cm
y-2
h
3m
6cm
y
A
36m2
54 cm2
y2 – 2y
P
30m
30m
4y-4
Find the area of the rectangle
3
5
AREA = 12
Group Practice
• Find the area of the figure. Consecutive
3
sides are perpendicular.
2
4
A = 114
2
units
5
6
5
Finding area? Ask these
questions…
1. What is the area formula for this shape?
2. What part of the formula do I already
have?
3. What part do I need to find?
4. How can I use a right triangle to find the
missing part?
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.
d1
d2
A = _________
d1∙d2
2
**WHAT DO YOU SEE
WITHIN THE DIAGRAM?
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
Finding area? Ask these
questions…
1. What is the area formula for this shape?
2. What part of the formula do I already
have?
3. What part do I need to find?
4. How can I use a right triangle to find the
missing part?
White Board Practice
A  12
5
5
6
•
Just talk about this one
White Board Practice
• Find the area of the figure
6
3
A9 3
3
60º
6
White Board Practice
• Find the area of the figure
A  30
12
5
13
•
Just talk about this one
White Board Practice
• Find the area of the figure
2
5
A  20
5
2
•
Just talk about this one
White Board Practice
• Find the area of the figure
– Side = 5cm
– 1 diagonal = 8cm
A  24
White Board Practice
• Find the area of the figure
A4 3
4
4
4
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
leg
median
base
leg
What type of trap
do we have if the
legs are
congruent?
Height
• The height of the trapezoid is the segment
that is perpendicular to the bases of the
trapezoid
b2
h
How do we measure
height for a trap?
Partners: Why is the
height perpendicular
to both bases?
b1
Theorem
The area of a trapezoid equals half the product
of the height and the sum of the bases.
b2
h
b1
demo
Labeling Height for Isosceles Trap
• Always label 2 heights when dealing with
an isosceles trap
White Board Practice
1. Find the area of the
trapezoid
7
5
A = 50
13
**talk**
White Board Practice
13
3. Find the area of the
trapezoid
14
9
A = 138
12
*talk*
Finding area? Ask these
questions…
1. What is the area formula for this shape?
2. What part of the formula do I already
have?
3. What part do I need to find?
4. How can I use a right triangle to find the
missing part?
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 2nd 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
side
Perimeter = sum
of sides
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?
s = length of side
p = 8s
r
a
s
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
r
a
x
3. Find the missing pieces
• What does ‘x’ represent?
Radius, Apothem, Perimeter
• Think 30-60-90
• Think 45-45-90
• Think SOHCAHTOA
r
A = ½ ap
a
p
A
8
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
r
A = ½ ap
a
8
p
4
A
24 3 48 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
IS THERE ANOTHER
AREA FORMULA FOR
THIS SHAPE?
r
a
p
A
A = ½ ap
5 2
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
r
A = ½ ap
5 2
r
x
a
a
p
5
40
A
100
IS THERE ANOTHER
AREA FORMULA FOR
THIS SHAPE?
r
A = ½ ap
a
p
A
8
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
a
x
r
A = ½ ap
a
8
p
A
4 3 48 96 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
a
x
r
a
p
A
A = ½ ap
24 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
a
x
r
A = ½ ap
4 3
a
p
6
A
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
   3.1415
d
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
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
For both formulas always
leave answers in

r
B
WHITEBOARDS
*put answers in terms of pi
r
d
C
A
15
8
26∏
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 tells us the
fraction or slice represents…
How much of the 360 ◦ of crust are we using from
our pizza?
A
C
60
B
Remember Circumference
The distance around the outside of a circle.
x◦
B
C
x◦
r
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
x◦
does it represent?
Circumference of circle
x◦
Degree measure of arc
LENGTH OF
ARC
x
(
)  2r
360
O
Example
If r = 6, what is the length of CB?
Measure of CB = 60◦
60 = 1
360
6
B
C
60◦
1 (2 ∙ 6) = 2
6
O
Remember Area
A  r
2
B
C
Sector of a circle
aka – the area of the piece
of pizza
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
2
(
)  r
360
Example
If r = 6, what is the area of sector
COB?
Measure of CB = 60◦
60 = 1
360
6
B
C
60◦
1 ( ∙ 62) = 6
6
O
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
3
5
1:2
3:5
a:b
1 to 2
3 to 5
a to b
a
b
Solving a Proportion
3 a

5 15
5a  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
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 a2:b2.
~
7
Area = 27
3
Scale Factor- 7: 3
Ratio of P – 7: 3
Ratio of A – 49 :9
WHITEBOARDS
OPEN BOOK TO PG. 458
•
•
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
–
•
(classroom exercises)
Ratio of P – 4:7
ANSWER # 13
a.
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 is the scale factor
• 2/3 = 12/x  x = 18
Remember
• Scale Factor a:b
• Ratio of perimeters a:b
• Ratio of areas a2:b2
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
random
experiment.
1. Total length of the line
2. Total area
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?
Event Space
P(e) 
Sample Space
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
Y
– 4 to 49
• Find the ratio of the areas
of WYV to quad WVZX
2
– 4 to 45
W
• Find the probability of a
point from the interior of
XYZ will lie in the
interior of quad XWYZ
V
5
– 45/49
X
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.
Test Review
Chapter Review
• 16
• 12
• 21
• 22
• Chapter test
–
–
–
–
4
9
12
15