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

© A Very Good Teacher 2007 Area of Com pos ite Figures 8, G.08A A Com pos ite 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 Com pos ite Figures, cont… Example: What is the area of the unshaded part of the rectangle below? 25 ft45 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 A = A - A - A = 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

© 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 ft = 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

© 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? 170˚ 110˚ 80˚ Meat Fruit Vegetables d=10, so r=5 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 square inches. What is the approximate length of BC? 8, G.08C A B C D 6 cm 9 cm 4.5 cm 6 cm 3 cm 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 = Bh V = (πr²)h V = (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 cm 2 cm 17 cm Left: A = ½bh A = ½∙9∙15 A = 67.5 A = bh A = 2∙17 A = 34 A = bh A = 2∙16 A = 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 in 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˚ 2 cm 3 cm 4 cm 6 cm The sides are in the ratio of 4 cm 6 cm The perimeter of the small ∆ is 10 cm The perimeter of the large ∆ is 15 cm 10 15

© 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 = ? 3x = 6∙10 3x = x = 20

© A Very Good Teacher 2007 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 A B C X Y Z 12 cm 19 cm 8 cm 16 cm 12∙XZ = 16∙8 12∙XZ = XZ = 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 in² A = 48 in² Ratio of Sides: Ratio of Perimeters: Ratio of Area:

© A Very Good Teacher 2007 Effects on Area, cont… The ratio of the sides is squared to find the ratio of the areas! Ratio of Sides SquaredRatio of Areas = If the ratio of sides is, what is the ratio of the areas? 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 Squared Ratio of Areas ? ? ?²?² ?²?² 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

© 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 CubedRatio of Volumes If the ratio of sides is, what is the ratio of the volumes? 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 Cubed Ratio of Volumes Answer: The new volume is 2 cubic centimeters