© A Very Good Teacher 2007 Exit Level TAKS Preparation Unit Objective 8.
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Transcript of © A Very Good Teacher 2007 Exit Level TAKS Preparation Unit Objective 8.
© A Very Good Teacher 2007
Exit Level
TAKS Preparation UnitObjective 8
© A Very Good Teacher 2007
Area of Composite Figures
8, G.08A
• A Composite Figure is made up different shapes
• Examples:
• To find the area: 1. Make a plan
2. Find the area of each part
3. Put each part back into the plan
© A Very Good Teacher 2007
Area of Composite Figures, cont…• Example: What is the area of the
unshaded part of the rectangle below?25 ft 45 ft
55 ft
95 ft
1. Make a Plan
2. Find the area of each part
3. Put each part back into the plan
A - A - A
A = l∙w
= 95∙55 = 5225
A = l∙w = 25∙25 = 625
A2
b h 45 55
2
2475
2 = 1237.5
A - A - A5225 625 1237.5 =3362.5 ft²
8, G.08A
© A Very Good Teacher 2007
Area of Sectors
• A Sector is a section of a circle like a pizza slice
• To find the Area of a Sector:– Find the area of the entire circle– Determine what portion of the circle in
contained in the sector
8, G.08B
360
x
2( )A r
© A Very Good Teacher 2007
Area of Sectors, cont… • Example: The shaded area in the circle
below represents the section of a playground used for tetherball. What is the approximate area of the section of the park used for tetherball?
8, G.08B
100˚
15 ft360
x
2( )A r
2 10015
360A
100225
360A = 196.35 ft²
© A Very Good Teacher 2007
Arc Length
• Arc Length is the distance around part of a circle (part of the circumference).
• To find the Arc Length:– Find the circumference of the circle– Determine what portion of the circle is
contained in the arc
8, G.08B
2360
xArc r
© A Very Good Teacher 2007
Arc Length, cont…
• Example: A paper plate with a 10 inch diameter is divided into three sections for different foods. What is the approximate length of the arc of the section containing vegetables?
2360
xArc r 170˚ 110˚
80˚
Meat
Fruit
Vegetables
d=10, so r=5
5236
110
0Arc
Arc Length = 9.6 in
8, G.08B
© A Very Good Teacher 2007
Using Pythagorean Theorem
• In order to use Pythagorean Theorem, you must have a right triangle!
• Example: The total area of trapezoid ABCD is 33.75 square inches. What is the approximate length of BC?
8, G.08C
2 2 2a b c
A B
CD
6 cm
9 cm
4.5
cm4
.5 c
m
6 cm
3 cm
2 2 2a b c 2 2 24.5 3 c
BC = 5.4
© A Very Good Teacher 2007
Volume of Solids• Identify the name of the Solid
– Cylinder, Rectangular Prism, Sphere, Cube, …
• Find the Formula on the Formula Chart!
8, G.08D
B is usually l∙w
© A Very Good Teacher 2007
Volume of Solids, cont…
• Example: Soda is packaged in cylindrical cans with the dimensions shown in the drawing. Find the approximate volume of this soda container.
8, G.08D
2.5 inches
4 inches
V = BhV = (πr²)hV = (π1.25²)4
V = 19.6 in³
© A Very Good Teacher 2007
Surface Area of Solids
• Identify the name of the Solid– Cylinder, Rectangular Prism, Sphere, Cube, …
• Find the Formula on the Formula Chart!
8, G.08D
Lateral means sides only (no top or bottom).
Be Careful! Most Surface Area Problems
Cannot be done by Formula!
© A Very Good Teacher 2007
Surface Area of Solids, cont…• Example: Adriana has a candy package shaped
like a triangular prism. The dimensions of the package are shown below. What is the surface area of the top, left, and right sides of the package?
8, G.08D
Top:
Right:
9 cm
15 c
m
2 cm
17 c
m
Left:
A = ½bhA = ½∙9∙15
A = 67.5
A = bhA = 2∙17
A = 34
A = bhA = 2∙16
A = 32
16 cm
= 133.5
© A Very Good Teacher 2007
Finding Similar Polygons ~• Similar polygons are the same shape, but
different sizes– Corresponding Angles are Congruent– Corresponding Sides are Proportional
• Examples:
4 in
6 in
4 in 6 in80˚ 80˚
80˚ 80˚2 cm
3 cm
4 cm
6 cm
8, G.11A
© A Very Good Teacher 2007
Similarity and Perimeter• When figures are similar, their perimeters
are also similar.
• Example:
8, G.11B
80˚ 80˚80˚ 80˚
2 cm3 cm
4 cm
6 cmThe sides are in the ratio of 2
34 cm 6 cm
The perimeter of the small ∆ is 10 cm
The perimeter of the large ∆ is 15 cm
10
15
2
3
© A Very Good Teacher 2007
Similarity and Perimeter, cont…
• Example: A rectangle has a length of 3 inches and a perimeter of 10 inches. What is the perimeter of a similar rectangle with a width of 6 inches?
8, G.11B
3 in
P = 10
6 in
P = ?
10
x
3
63x = 6∙10
3x = 603 3
x = 20
© A Very Good Teacher 2007
12 8
16
BX
YZ XZ
Solving Problems with Similar Figures
• Use RATIOS
• Example: Look at the figures below. If , which is closest to the length of XZ?
8, G.11C
ABC XYZ
A B
C
XY
Z12 cm
19 cm8 cm
16 cm
AB BC AC
XY YZ XZ
12∙XZ = 16∙812∙XZ = 128
12 8
16 XZ
12 12XZ = 10.67
© A Very Good Teacher 2007
Effects on Area• When similar figures are enlarged, the
area changes, but not in the same ratio as the perimeter
• Let’s take a look:
8, G.11D
3 in
6 in
4 in
8 in
A = 12 in²A = 48 in²
Ratio of Sides:
Ratio of Perimeters:
Ratio of Area:
3 1
6 2
14 1
28 2
12 1
48 4
© A Very Good Teacher 2007
Effects on Area, cont…• The ratio of the sides is squared to find
the ratio of the areas!
2
2
1
2
1
4
1
2
Ratio of Sides
Squared Ratio of Areas
=
If the ratio of sides is , what is the ratio of the areas?
2
3
8, G.11D
© A Very Good Teacher 2007
Using Effects on Area• Example: If the surface area of a cube is
increased by a factor of 16, what is the change in the length of the sides of the cube?
8, G.11D
Ratio of Sides
SquaredRatio of Areas
??
?²?²
161
4
1
Answer: The length is 4 times the original length
© A Very Good Teacher 2007
Effects on Volume
• How does the change is sides effect the Volume of a solid?
8, G.11D
12 cm
16 cm
8 cm
18 cm
24 cm
12 cm
V = 8∙12∙16
V = 12∙18∙24
V =1536
V = 5184
Ratio of Sides
Ratio of Volumes
2
3
8
12
1536
51848
27
© A Very Good Teacher 2007
Effects on Volume, cont…• The ratio of the sides is cubed to find the
ratio of the volumes!
Ratio of Sides
Cubed Ratio of Volumes
8
27
3
3
2
3
2
3
If the ratio of sides is , what is the ratio of the volumes?
2
5
8, G.11D
© A Very Good Teacher 2007
Using Effects on Volume• Example: A rectangular solid has a volume
of 54 cubic centimeters. If the length, width, and height are all changed to 1/3 their original size, what will be the new volume of the rectangular solid?
8, G.11D
Ratio of Sides
CubedRatio of Volumes
1
27
3
3
1
3
1
3 2x
27 54x
1
27 54
x
Answer: The new volume is 2 cubic centimeters