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Geometry Concepts for TAKS Preparation

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Prepare for TAKS with area calculations for composite figures, sectors, arc lengths, Pythagorean theorem, and volume calculations for solids. Includes examples and formulas.

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Geometry Concepts for TAKS Preparation

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  1. Exit Level TAKS Preparation Unit Objective 8

  2. Area of Composite Figures • A Composite Figure is made up different shapes • Examples: • To find the area: • Make a plan • Find the area of each part • Put each part back into the plan 8, Ge1A

  3. 25 ft 45 ft A - A - A A - A - A 55 ft A = A A 95 ft Area of Composite Figures, cont… • Example: What is the area of the unshaded part of the rectangle below? • Make a Plan 2. Find the area of each part = 95∙55 = 5225 l∙w = 625 = l∙w = 25∙25 = 1237.5 3. Put each part back into the plan =3362.5 ft² 5225 1237.5 625 8, Ge1A

  4. 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, Ge1B

  5. 100˚ 15 ft 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? = 196.35 ft² 8, Ge1B

  6. ArcLength • ArcLength is the distance around part of a circle (part of the circumference). • To find the ArcLength: • Find the circumference of the circle • Determine what portion of the circle is contained in the arc 8, Ge1B

  7. Vegetables Meat 110˚ 170˚ 80˚ Fruit ArcLength, cont… d=10, so r=5 • 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? Arc Length = 9.6 in 8, Ge1B

  8. 6 cm A B 4.5 cm C D 9 cm 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? 6 cm 4.5 cm 3 cm BC = 5.4 8, Ge1C

  9. Volume of Solids • Identify the name of the Solid • Cylinder, Rectangular Prism, Sphere, Cube, … • Find the Formula on the Formula Chart! B is usually l∙w 8, Ge1D

  10. 2.5 inches 4 inches 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. V = Bh V = (πr²)h V = (π1.25²)4 V = 19.6 in³ 8, Ge1D

  11. Surface Area of Solids Be Careful! Most Surface Area Problems Cannot be done by Formula! • Identify the name of the Solid • Cylinder, Rectangular Prism, Sphere, Cube, … • Find the Formula on the Formula Chart! Lateral means sides only (no top or bottom). 8, Ge1D

  12. 9 cm 2 cm 15 cm 17 cm 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? Top: A = ½bh A = 67.5 A = ½∙9∙15 Right: A = bh A = 34 A = 2∙17 16 cm Left: A = bh A = 32 A = 2∙16 = 133.5 8, Ge1D

  13. 6 cm 4 in 4 cm 6 in 4 in 80˚ 80˚ 6 in 80˚ 80˚ 2 cm 3 cm Finding Similar Polygons ~ • Similar polygons are the same shape, but different sizes • Corresponding Angles are Congruent • Corresponding Sides are Proportional • Examples: 8, Gf1A

  14. 6 cm 4 cm 80˚ 80˚ 80˚ 80˚ 2 cm 3 cm Similarity and Perimeter • When figures are similar, their perimeters are also similar. • Example: The sides are in the ratio of 6 cm 4 cm 10 The perimeter of the small ∆ is 10 cm The perimeter of the large ∆ is 15 cm 15 8, Gf1B

  15. 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? P = 10 3x = 6∙10 P = ? x = 20 3 in 3x = 60 3 3 6 in 8, Gf1B

  16. 12 cm B A Z 8 cm 19 cm C X Y 16 cm Solving Problems with Similar Figures • Use RATIOS • Example: Look at the figures below. If , which is closest to the length of XZ? 12∙XZ = 16∙8 XZ = 10.67 12∙XZ = 128 12 12 8, Gf1C

  17. 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: Ratio of Sides: Ratio of Perimeters: A = 12 in² 4 in A = 48 in² 8 in 3 in Ratio of Area: 6 in 8, Gf1D

  18. Effects on Area, cont… • The ratio of the sides is squared to find the ratio of the areas! Ratio of Sides Squared Ratio of Areas = If the ratio of sides is , what is the ratio of the areas? 8, Gf1D

  19. ? ?² 16 ? ?² 1 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? Ratio of Sides Ratio of Areas Squared 4 1 Answer: The length is 4 times the original length 8, Gf1D

  20. Effects on Volume • How does the change is sides effect the Volume of a solid? Ratio of Sides 8 cm V = 8∙12∙16 12 cm V =1536 16 cm Ratio of Volumes 12 cm V = 12∙18∙24 18 cm V = 5184 24 cm 8, Gf1D

  21. 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 If the ratio of sides is , what is the ratio of the volumes? 8, Gf1D

  22. 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? Ratio of Volumes Ratio of Sides Cubed Answer: The new volume is 2 cubic centimeters 8, Gf1D

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