Chapter 11 Areas of Plane Figures Understand what is meant by the area of a polygon. Know and use...
Transcript of Chapter 11 Areas of Plane Figures Understand what is meant by the area of a polygon. Know and use...
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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.
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11-1: Area of Rectangles
Objectives
• Learn and apply the area formula for a square and a rectangle.
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Area
A measurement of the region covered by a geometric figure and its interior.
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Theorem
The area of a rectangle is the product of the base and height.
b
hArea = b x h
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Base
• Any side of a rectangle or other parallelogram can be considered to be a base.
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Altitude
• Altitude to a base is any segment perpendicular to the line containing the base from any point on the opposite side.
• Called Height
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Postulate
The area of a square is the length of the side squared.
s
s
Area = s2
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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.
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Find the area
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Area Addition Postulate
The area of a region is the sum of the areas of its non-overlapping parts
A
B
C
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Remote Time
Classify each statement as True or False
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Question 1
• If two figures have the same areas, then they must be congruent.
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Question 2
• If two figures have the same perimeter, then they must have the same area.
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Question 3
• If two figures are congruent, then they must have the same area.
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Question 4
• Every square is a rectangle.
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Question 5
• Every rectangle is a square.
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Question 6
• The base of a rectangle can be any side of the rectangle.
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White Board Practice
b 12m 9cm y-2
h 3m y
A 54 cm2
b
h
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White Board Practice
b 12m 9cm y-2
h 3m 6cm y
A 36m2 54 cm2 y2 – 2y
b
h
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Group Practice
• Find the area of the figure. Consecutive sides are perpendicular.
54
23
6
5
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Group Practice
• Find the area of the figure. Consecutive sides are perpendicular.
54
23
6
5
A = 114
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11-2: Areas of Parallelograms, Triangles, and Rhombuses
Objectives
• Determine and apply the area formula for a parallelogram, triangle and rhombus.
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Tons of formulas to memorize
• So don’t memorize them
• Understand them !
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Refresh my memory…
What is the area of a rectangle ?
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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.
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h
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h
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h
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h
Cut Slide over and tape
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h
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Do we have any leftover paper?
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So this means the area must be the same
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Theorem
The area of a parallelogram is the product of the base times the height to that base.
b
h
Area = b x h
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But Wait….
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What do we have ?
X 2
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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
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Theorem
The area of a rhombus equals half the product of the diagonals.
d1
d2
1 2
1
2Area d d
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Remote Time
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White Board Practice
• Find the area of the figure
4
4
4
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White Board Practice
• Find the area of the figure
4
4
4
34A
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White Board Practice
• Find the area of the figure
3
6
6
3
60º
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White Board Practice
• Find the area of the figure
3
6
6 39A
3
60º
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White Board Practice
• Find the area of the figure
5 5
6
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White Board Practice
• Find the area of the figure
5 5
12A
6
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White Board Practice
• Find the area of the figure
2
25
5
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White Board Practice
• Find the area of the figure
20A2
25
5
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White Board Practice
• Find the area of the figure
4
5
4
55
5
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White Board Practice
• Find the area of the figure
24A4
5
4
55
5
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White Board Practice
• Find the area of the figure
12
13
5
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White Board Practice
• Find the area of the figure
30A12
13
5
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11-3: Areas of Trapezoids
Objectives
• Define and apply the area formula for a trapezoid.
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Trapezoid Review
A quadrilateral with exactly one pair of parallel sides.
base
base
leg leg
height
median
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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
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Height
• The height of the trapezoid is the segment that is perpendicular to the bases of the trapezoid
b1
h
b2
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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
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White Board Practice
13
5
71. Find the area of the trapezoid and the length of the median
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White Board Practice
13
5
71. Find the area of the trapezoid and the length of the median
A = 50Median = 10
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White Board Practice
13
6
52. Find the area of the trapezoid and the length of the median
10
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White Board Practice
13
6
52. Find the area of the trapezoid and the length of the median
10
A = 54Median = 9
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White Board Practice3. Find the area of the trapezoid and the length of the median
14
12
13
9
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White Board Practice3. Find the area of the trapezoid and the length of the median
14
12
13
9
A = 138Median = 11.5
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Group Practice
• A trapezoid has an area of 75 cm2 and a height of 5 cm. How long is the median?
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Group Practice
• A trapezoid has an area of 75 cm2 and a height of 5 cm. How long is the median?
Median = 5 cm
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Group Practice
• Find the area of the trapezoid
8 8
8
60º
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Group Practice
• Find the area of the trapezoid
8 8
8
60º
Area = 348
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Group Practice
• Find the area of the trapezoid
45º
23
4
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Group Practice
• Find the area of the trapezoid
Area = 2
33
45º
23
4
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Group Practice
• Find the area of the trapezoid
12
30
30º 30º
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Group Practice
• Find the area of the trapezoid
Area = 363
12
30
30º 30º
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11.4 Areas of Regular Polygons
Objectives
• Determine the area of a regular polygon.
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Regular Polygon Review
side
All sides congruentAll angles congruent
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Center of a regular polygon
center
is the center of the circumscribed circle
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Radius of a regular polygon
center
is the radius of the circumscribed circle
is the distance fromthe center to a vertex
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Central angle of a regular polygon
Central angle
Is an angle formed bytwo radii drawn to consecutive vertices
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Apothem of a regular polygon
apothem
the perpendicular distance from the center to a side of the polygon
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Regular Polygon Review
side
center
radiusapothem
central angle
Perimeter = sum of sides
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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
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TheoremThe area of a regular polygon is half the product of
the apothem and the perimeter.
a
s
r
P = 8s
1
2Area ap
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RAPA
• R adius
• A pothem
• P erimeter
• A rea
a
s
r
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Radius, Apothem, Perimeter
1. Find the central angle 360 n
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Radius, Apothem, Perimeter
2. Divide the isosceles triangle into two congruent right triangles
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Radius, Apothem, Perimeter
ra
3. Find the missing piecesx
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Radius, Apothem, Perimeter
• Think 30-60-90
• Think 45-45-90
• Thing SOHCAHTOA
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r a p A
ra
x
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
25A = ½ ap
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r a p A
5 40 100
ra
x
25A = ½ ap
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r a p A
ra
x
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
3A = ½ ap
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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
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r a p A
8
ra
x
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap
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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
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r a p A
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap36
ra
x
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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
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r a p A
8
r a
x
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap
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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
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r a p A
1. Central angle2. ½ of central angle3. 45-45-90
30-60-90SOHCAHTOA
A = ½ ap324
r a
x
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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
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11.5 Circumference and Areas of Circles
Objectives
• Determine the circumference and area of a circle.
3.1415C
d
r
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r a p A ??
Doesn’t work! Why?
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r a p A ??
Doesn’t work! Why?
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CircumferenceThe distance around the outside of a circle.
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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.
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Experiment
6. Make a ratio of the Circumference.
Diameter
7. Give the ratio in decimal form to the nearest hundreth.
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ExperimentCircle Number
Circumference (nearest mm)
Diameter (nearest mm)
Ratio of Circumference/Diameter (as a decimal)
1
2
3
4
5
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What do you think?
1. How does the measurement of the circumference compare to the measurement of the diameter?
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2. Were there any differences in results? If so, what were they?
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3. Did you recognize a pattern? Were you able to verify a pattern?
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• 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
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CircumferenceThe distance around the outside of a circle.
2C d r r
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Area
B
The area of a circle is the product of pi times the square of the radius.
2A rr
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11.6 Arc Length and Areas of Sectors
Objectives
• Solve problems about arc length and sector and segment area.
r
A
B
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Remember Circumference
C = 2r
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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
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Remember Area ?
2rA
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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
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White Board Practice
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11-7 Ratios of Areas
Objectives
• Solve problems about the ratios of areas of geometric figures.
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Comparing Areas of Triangles
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Two triangles with equal heights
4 4
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Two triangles with equal heights
bhA2
1
4 4
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Two triangles with equal heights
bhA2
1
4 4
7 3
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Ratio of their areas
4 4
7 3
6
14
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Ratio of areas = ?
4 4
7 3
3
7
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If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.
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Two triangles with equal bases
5 5
8
2
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Ratio of Areas
5 5
8
2
5
20
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Ratio of Areas = ?
5 5
8
2
1
4
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If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights.
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If two triangles are similar, then the ratio of their areas equals the square of their scale factor.
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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.
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White Board Practice
• Find the ratio of the areas of ABC: ADB
A
B
CD
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White Board Practice
• Find the ratio of the areas of ABD: BCD
A
B
CD
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Remember
• Scale Factor a:b
• Ratio of perimeters a:b
• Ratio of areas a2:b2
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Remote Time
• True or False
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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
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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
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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
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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
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11-8: Geometric Probability
Solve problems aboutGeometric probability.
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Event: A possible outcome in a random experiment.
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Sample SpaceThe number of all possible outcomes in a random experiment.
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Probability•The calculation of the possible outcomes in a random experiment.
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( )
Event SpaceP e
Sample Space
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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.
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Homework Set 11.8
Pg 463
(2-10 even
Pg 465 (1-10)