© A Very Good Teacher 2007 Exit Level TAKS Preparation Unit Objective 8

© A Very Good Teacher 2007

Exit Level

TAKS Preparation UnitObjective 8


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


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


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


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


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


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


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


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


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


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!


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


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


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


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


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


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


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


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


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


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


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

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© A Very Good Teacher 2007 Exit Level TAKS Preparation Unit Objective 8