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.
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.
h
Area = b x h
b
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.
Area = s2
s
s
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.
Find the area
Area Addition Postulate
The area of a region is the sum of the areas of its
non-overlapping parts
B
A
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
h
b
b
12m
h
3m
A
9cm
y-2
y
54 cm2
White Board Practice
h
b
b
12m
9cm
y-2
h
3m
6cm
y
A
36m2
54 cm2
y2 – 2y
Group Practice
• Find the area of the figure. Consecutive
sides are perpendicular.
3
2
4
5
6
5
Group Practice
• Find the area of the figure. Consecutive
sides are perpendicular.
3
2
4
A = 114
5
6
5
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
h
h
h
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.
h
Area = b x h
b
But Wait….
What do we have ?
X2
Theorem
The area of a triangle equals half the product
of the base times the height to that base.
1
Area  b  h
2
h
b
Theorem
The area of a rhombus equals half the product
of the diagonals.
d1
d2
1
Area  d1  d 2
2
Remote Time
White Board Practice
• Find the area of the figure
4
4
4
White Board Practice
• Find the area of the figure
A4 3
4
4
4
White Board Practice
• Find the area of the figure
6
3
3
60º
6
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
5
5
6
White Board Practice
• Find the area of the figure
5
5
6
A  12
White Board Practice
• Find the area of the figure
2
5
5
2
White Board Practice
• Find the area of the figure
2
5
5
2
A  20
White Board Practice
• Find the area of the figure
5
4
5
5
4
5
White Board Practice
• Find the area of the figure
5
4
5
5
4
5
A  24
White Board Practice
• Find the area of the figure
12
5
13
White Board Practice
• Find the area of the figure
12
5
13
A  30
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
height
base
leg
Median
• Remember the median
is the segment that
connects the midpoints
of the legs of a
trapezoid.
• Length of median
= ½ (b1+b2)
b2
median
b1
Height
• The height of the trapezoid is the segment
that is perpendicular to the bases of the
trapezoid
b
2
h
b1
Theorem
The area of a trapezoid equals half the product
of the height and the sum of the bases.
b2
1
Area  h(b1  b2 )
2
h
b1
Sketch
White Board Practice
1. Find the area of the
trapezoid and the length
of the median
7
5
13
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
2. Find the area of the
trapezoid and the length
of the median
5
10
6
13
White Board Practice
2. Find the area of the
trapezoid and the length
of the median
5
10
6
A = 54
Median = 9
13
White Board Practice
3. Find the area of the
trapezoid and the length
of the median
13
14
9
12
White Board Practice
3. Find the area of the
trapezoid and the length
of the median
13
14
9
A = 138
Median = 11.5
12
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
• 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
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
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
side
Center of a regular polygon
is the center of the
circumscribed circle
center
Radius of a regular polygon
is the radius of the
circumscribed circle
center
is the distance from
the center to a vertex
Central angle of a regular polygon
Is an angle formed by
two radii drawn to
consecutive vertices
Central angle
Apothem of a regular polygon
the perpendicular
distance from the
center to a side
of the polygon
apothem
Regular Polygon Review
central angle
center
apothem
side
Perimeter = sum of sides
Polygon Review Continued
Sides
3
4
5
6
7
8
n
Name
1 interior 
Triangle
60
Square
90
Pentagon
108
Hexagon
120
Septagon
128.6
Octagon
135
n-gon
(n-2)180
n
1 Central 
120
90
72
60
51.4
45
360
n
Theorem
The area of a regular polygon is half the
product of the apothem and the perimeter.
P = 8s
r
a
s
1
Area  ap
2
RAPA
•
•
•
•
R adius
A pothem
P erimeter
A rea
r
a
s
Radius, Apothem, Perimeter
1. Find the central angle 360
n
Radius, Apothem, Perimeter
2. Divide the isosceles triangle into two congruent right triangles
Radius, Apothem, Perimeter
r
a
x
3. Find the missing pieces
Radius, Apothem, Perimeter
• Think 30-60-90
• Think 45-45-90
• Thing SOHCAHTOA
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
r
a
p
A
A = ½ ap
3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
r
a
p
A
A = ½ ap
6
3 8 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
12
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
r
a
p
A
A = ½ ap
6 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
r
A = ½ ap
a
2
p
1
A
6 3 3 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
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
r a p A ??
Doesn’t work! Why?
r a p A ??
Doesn’t work! Why?
Circumference
The distance around the outside of a circle.
Experiment
1. Select 5 circular objects
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 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.
Experiment
Circle Circumference Diameter
Number (nearest mm) (nearest mm)
1
2
3
4
5
Ratio of
Circumference/Diameter
(as a decimal)
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
Circumference
The distance around the outside of a circle.
r
C   d  2 r
Area
The area of a circle is the product of pi times
the square of the radius.
r
A r
2
B
11.6 Arc Length and Areas of
Sectors
Objectives
• Solve problems about arc length and sector
and segment area.
A
r
B
Remember Circumference
C = 2r
Arc Length
The length of an arc is the product of the
circumference of the circle and the ratio of
the circle that the arc represents.
B
C
x
LengthBC 
2r
360
x
O
r
Remember Area ?
A  r
2
Sector Area
The 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.
x
2
AreaofSectorBC 
r
360
C
B
r
x
A
White Board Practice
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
Two triangles with equal heights
1
A  bh
2
4
4
Two triangles with equal heights
1
A  bh
2
4
7
4
3
Ratio of their areas
4
7
14
6
4
3
Ratio of areas = ?
4
7
7
3
4
3
If two triangles have
equal heights, then the
ratio of their areas
equals the ratio of their
bases.
Two triangles with equal bases
8
2
5
5
Ratio of Areas
8
20
5
2
5
5
Ratio of Areas = ?
8
4
1
2
5
5
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.
Theorem
If 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
2
2
a :b .
White Board Practice
• Find the ratio of the areas of ABC: ADB
B
A
D
C
White Board Practice
• Find the ratio of the areas of ABD: BCD
B
A
D
C
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 3 : 2
11-8: Geometric
Probability
Solve problems
aboutGeometric
probability.
Event:
A possible
outcome in a
random
experiment.
Sample Space
The number of
all possible
outcomes in a
random
experiment.
Probability
•The calculation of
the possible
outcomes in a
random experiment.
Event Space
P (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)