Chapter 11 Areas of Plane Figures
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Transcript of 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
White Board Practice
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
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
White Board Practice
• Find the area of the figure
4
4
4
White Board Practice
• Find the area of the figure
4
4
4
34A
White Board Practice
• Find the area of the figure
3
6
6
3
60º
White Board Practice
• Find the area of the figure
3
6
6 39A
3
60º
White Board Practice
• Find the area of the figure
5 5
6
White Board Practice
• Find the area of the figure
5 5
12A
6
White Board Practice
• Find the area of the figure
2
25
5
White Board Practice
• Find the area of the figure
20A2
25
5
White Board Practice
• Find the area of the figure
4
5
4
55
5
White Board Practice
• Find the area of the figure
24A4
5
4
55
5
White Board Practice
• Find the area of the figure
12
13
5
White Board Practice
• Find the area of the figure
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
White Board Practice
13
5
71. Find the area of the trapezoid and the length of the median
White Board Practice
13
5
71. Find the area of the trapezoid and the length of the median
A = 50Median = 10
White Board Practice
13
6
52. Find the area of the trapezoid and the length of the median
10
White Board Practice
13
6
52. Find the area of the trapezoid and the length of the median
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
• Find the area of the trapezoid
8 8
8
60º
Area = 348
Group Practice
• Find the area of the trapezoid
45º
23
4
Group Practice
• Find the area of the trapezoid
Area = 2
33
45º
23
4
Group Practice
• Find the area of the trapezoid
12
30
30º 30º
Group Practice
• Find the area of the trapezoid
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
Radius, Apothem, Perimeter
2. Divide the isosceles triangle into two congruent right triangles
Radius, Apothem, Perimeter
ra
3. Find the missing piecesx
Radius, Apothem, Perimeter
• 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
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
3A = ½ ap
r a p A
12
ra
x
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
3A = ½ ap
6 38
r a p A
8
ra
x
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap
r a p A
8 4
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap324
ra
x
348
r a p A
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap36
ra
x
r a p A
2 1
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap36
ra
x
33
r a p A
8
r a
x
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap
r a p A
8 48
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
34A = ½ ap
396
r a
x
r a p A
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap324
r a
x
r a p A
6
1. Central angle2. ½ of central angle3. 45-45-90
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
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
bhA2
1
4 4
Two triangles with equal heights
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.
White Board Practice
• Find the ratio of the areas of ABC: ADB
A
B
CD
White Board Practice
• 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)