Suggested Pacing. Day 3 8.7B. Rectangular prisms, Triangular Prisms, and Cylinders Day 8 CRM 6 Assessment

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1 Surface Area Support Materials Day 1 8.7B Surface Area of Rectangular prisms, Triangular Prisms, and Cylinders Day 6 Solving Problem Situations with Surface Area and Volume Suggested Pacing *Adjust as needed to meet the needs of your students Day 2 8.7B Surface Area of Rectangular prisms, Triangular Prisms, and Cylinders Day 7 Solving Problem Situations with Surface Area and Volume Day 3 8.7B Surface Area of Rectangular prisms, Triangular Prisms, and Cylinders Day 8 CRM 6 Assessment Day 4 8.7B Determining when to use Lateral or Total Surface Area Day 5 8.7B Solving Problem Situations with Surface Area and Volume Table of Contents Activity Name TEKS Description Investigation Activity And That s A Wrap Foldable Activity Making Connections Carnegie Lesson 17.2 Textbook Assignments Skill Practice More Foldable Ideas Surface Area Rectangular Prism 8.7B 8.7B 8.7B 8.7B Students will investigate surface area by using concrete 3-D shapes. Students will calculate surface area of rectangular prisms, triangular prisms and cylinders using nets and connect actions to the formulas on the STAAR 8 Reference chart. This lesson focuses on using nets to calculate the surface area of a cylinder. These are links to additional foldable ideas for anchors and interactive notebooks. Austin ISD Middle School Math

2 Surface Area Support Materials Table of Contents Activity Name TEKS Description Triangular Prism Supporting Activity Surface Area Sort 8.7B Students will sort problems situations into the categories of lateral surface area or total surface area and then solve. Carnegie Lesson 17.5 Textbook Assignments Skill Practice 8.7A 8.7B Surface Area Go Around 8.7B Engaging Math 8.7B This lesson provides real-world problems to solve using the volume and surface area formulas for cylinders, cones, and spheres. Use these 8 problems for stations, additional practice, or a class game. Focus activities are: Figure Math, Totally Tubular Text Message: Roll-O-Candy, and Find Someone Who These activities are good for extra practice and quick interventions. Austin ISD Middle School Math

3 Volume Disco balls are spheres that reflect light in all different directions. They were really popular in dance clubs throughout the 1960s, 1970s, and 1980s Drum Roll, Please! Volume of a Cylinder Scratching the Surface Surface Area of a Cylinder Piling On! Volume of a Cone All Bubbly Volume of a Sphere Practice Makes Perfect Volume and Surface Area Problems

4 Chapter 17 Overview This chapter develops formulas for the volumes of cones, right circular cylinders, and spheres. Volume formulas are used to solve real-world problems. Lessons TEKS Pacing Highlights Models Worked Examples Peer Analysis Talk the Talk Technology 17.1 Volume of a Cylinder 8.2.B 8.6.A 8.7.A 1 This lesson reviews the characteristics of a cylinder, and then focuses on using unit cubes and estimation to calculate the volume of a cylinder. Questions ask students to investigate the effect doubling the length of the radius of the base of the cylinder has on the volume of the cylinder. X X 17.2 Surface Area of a Cylinder 8.7.B 1 This lesson focuses on using nets to calculate the surface area of a cylinder. X X 17.3 Volume of a Cone 8.6.B 8.7.A 1 This lesson investigates the volume of a cone using nets for a cylinder, cone, and birdseed. Questions ask students to write a formula for the volume of a cone, and use the formula to solve real-world problems. X 17.4 Volume of a Sphere 8.7.A 1 This lesson provides the formula for the volume of a sphere. Questions ask students to calculate the volume of various real-world objects. X 17.5 Volume and Surface Area Problems 8.6.A 8.7.A 8.7.B 1 This lesson provides real-world problems to solve using the volume and surface area formulas for cylinders, cones, and spheres. X X 925A Chapter 17 Volume

5 Skills Practice Correlation for Chapter 17 Lesson Problem Set Objective(s) Vocabulary Volume of a Cylinder Surface Area of a Cylinder 1 6 Identify radius, diameter, and height of cylinders 7 12 Calculate the area of bases then determine the volume of cylinders Calculate volume of cylinders Vocabulary 1 6 Calculate area of rectangles in nets of cylinders 7 14 Calculate surface areas of cylinders Vocabulary 17.3 Volume of a Cone 1 8 Identify radius, diameter, and height of cones 9 18 Calculate volume of cones Use information from one cone to determine information for another cone 17.4 Volume of a Sphere 1 6 Vocabulary Identify radius, diameter, and distance from center then determine the approximate circumference of spheres 17.5 Volume and Surface Area Problems 7 16 Calculate volume of spheres Calculate volume of spheres using information given 1 6 Identify solids and measurements determined by formulas 7 13 Use volume formulas to solve problems Chapter 17 Volume 925B

6 Scratching the Surface Surface Area of a Cylinder Learning Goals In this lesson, you will: Determine the surface area of cylinders. Key Terms surface area of a cylinder Essential Ideas The bases of a cylinder are two congruent circles and their interiors. The radius of a cylinder is the radius of a base. The formula for the surface area S of a cylinder is S 5 2pr 2 1 2prh. Texas Essential Knowledge and Skills for Mathematics Grade 8 (7) Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to: (B) use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders 17.2 Surface Area of a Cylinder 941A

7 Overview Students compare the volume of two different cylinders made from the same sheet of poster board. Students use nets to construct models of cylinders, and use their nets to write a formula for the surface area of a cylinder. Students use the formula to calculate the surface areas of cylinders. 941B Chapter 17 Volume

8 Warm Up Consider the rectangular prism shown. 3 in. 10 in. 4 in. 1. Identify the length, width, and height of the prism. The length is 10 inches, the width is 4 inches, and the height is 3 inches. 2. Draw a net of the prism. Label the dimensions. 10 in. 4 in. 4 in. 3 in. 4 in. 4 in. 3 in. 3. Calculate the volume of the prism. V 5 l 3 w 3 h cubic inches 4. Calculate the surface area of the prism. S 5 2(10 3 4) 1 2(4 3 3) 1 2(10 3 3) 5 2(40) 1 2(12) 1 2(30) square inches 17.2 Surface Area of a Cylinder 941C

9 941D Chapter 17 Volume

10 Scratching the Surface Surface Area of a Cylinder Learning Goals In this lesson, you will: Determine the surface area of cylinders. Key Terms surface area of a cylinder Archimedes of Syracuse, Sicily (c BC), was an ancient Greek mathematician, physicist, and engineer. Among other things, Archimedes discovered formulas involving cylinders. Archimedes considered his greatest achievement to be his discovery of the relationship between a cylinder and another shape which you will learn about later in this chapter. Archimedes must have really liked cylinders, because he even asked for a sculpted cylinder to be placed on his tomb! In this lesson, you will determine a formula for the surface area of a cylinder. Think about the pieces that make up a net of a cylinder. How do you think the formula for the surface area of a cylinder will be different than the formula for the volume of a cylinder? 17.2 Surface Area of a Cylinder 941

11 Problem 1 A standard piece of poster board is used to make a cylinder using tape. The dimensions of the poster board are given and students use the information to compare the volume of two possible cylinders. Problem 1 Poster Board Grouping Have students complete the problem with a partner. Then share the results as a class. 36 in. Share Phase, Question 1 How many different cylinders can be made from taping together the ends of the rectangular poster board? To determine the volume of each cylinder, what formula is used? What unit of measure is used when describing the volume of each cylinder? Why do you suppose two different volumes are possible, when the cylinders are made from the same piece of poster board? Does this make sense? 24 in. A standard piece of poster board is 24 inches by 36 inches. A cylindrical shape can be made by taping two ends of the poster board together. 1. Explain which way the poster board should be taped to result in the cylinder having the greatest volume. Justify your conclusion. C 5 2πr C 5 2πr πr πr 12 5 πr 18 5 πr r 3.8 r 5.7 V 5 πr 2 h V 5 πr 2 h V 5 π(3.8) 2 (36) V 5 π(5.7) 2 (24) V cubic inches V cubic inches The poster board should be rolled and taped such that the 24-inch edges are taped together. 942 Chapter 17 Volume

12 Problem 2 Students will use one of the nets from Question 1 to construct a model. Students will note that the length of the rectangle is the circumference of a circular base, which is needed to calculate the surface area. Problem 2 Surface Area of a Cylinder Recall that the surface area of a solid three-dimensional object is the total area of the outside surfaces (faces and bases) of the solid. Surface area is described using square units of measure. It follows that the surface area of a cylinder is the total area of the outside surfaces of the cylinder. 1. Clarissa, Xavier, and Umi each drew a different net, as shown. Clarissa s Net Xavier s Net Be sure English Language Learners have a full understanding of surface area and volume. Have students create concept maps describing circumference and area of circles and surface area and volume of cylinders. Encourage them to write sentences that make comparisons such as, Circumference and surface area measure the outside; area and volume measure the inside. Check to see that students make valid comparisons and have a firm grasp of dimensions, formulas, and units for each form of measure. Umi s Net Whose net represents the net of a cylinder? Explain your reasoning. Each student s net represents the net of a cylinder. Even though the positioning of the circular bases is different, when each net is folded, the resulting solid is the same. Grouping Have students complete Questions 1 through 8 with a partner. Then share the results as a class. Share Phase, Question 1 How is the surface area of a cylinder different from the volume of a cylinder? How are the three students nets similar? How are they different? 17.2 Surface Area of a Cylinder 943

13 Share Phase, Questions 2 and 3 Why did you choose the net that you chose? Explain the units of measure you used for each part in Question On another piece of paper, sketch a larger version of one of the nets from the previous question. Then cut out the net and tape the longer sides together to form a cylinder. a. What shapes were used to form your model of a cylinder? The model of a cylinder is formed from a rectangle and two circles. b. Use your model to help you explain how you would calculate the surface area of a cylinder. The surface area of a cylinder is the sum of the areas of the circles and the area of the rectangle. 3. Remove the tape from your model of a cylinder. a. Calculate the length of the rectangle. Answers will vary. Use a ruler to calculate your measurements. b. How does the width of the rectangle relate to the circle? The width of the rectangle is the same as the circumference of the circle. c. Calculate the width of the rectangle. Answers will vary. d. Calculate the area of the rectangle. Answers will vary. 944 Chapter 17 Volume

14 Share Phase, Questions 4 through 8 Are your measurements the same or different from other students in your class? How is the lateral surface area of a cylinder different from the surface area of a cylinder? What other ways could you write the formula for the surface area of a cylinder? 4. Calculate the area of the bases of your model of a cylinder. Use 3.14 for π. Answers will vary. 5. Calculate the lateral surface area of your model of a cylinder. Use 3.14 for π. Answers will vary. Recall that the lateral surface area of a solid three-dimensional object is the sum of the surface areas of all its faces excluding the bases of the solid. 6. Write a formula for the lateral surface area of a cylinder. Use L for the lateral surface area, r for the radius, h for the height, and π for pi. L 5 2πrh 7. Calculate the surface area of your model of a cylinder. Answers will vary. 8. Write a formula for the surface area of a cylinder. Use S for the surface area, r for the radius, h for the height, and π for pi. S 5 2πr 2 1 2πrh 17.2 Surface Area of a Cylinder 945

15 Grouping Have students complete Questions 9 and 10 with a partner. Then share the results as a class. Share Phase, Questions 9 and 10 Describe which units you used and why. Compare the lateral surface areas and total surface areas of the cylinders in Question 9. Will one always be greater than the other? Explain. 9. Calculate the lateral surface area and the total surface area of each cylinder. Use 3.14 for π. Round your answer to the nearest whole unit of measure if necessary. a. 6 inches 20 inches S 5 2π(6 2 ) 1 2π(6)(20) 5 72π 1 240π 5 312π The surface area of the cylinder is about 980 square inches. r 5 6; h 5 20 L 5 2π(6)(20) 5 240π The lateral surface area of the cylinder is about 754 square inches. b. 6 centimeters S 5 2π(3 2 ) 1 2π(3)(4) 5 18π 1 24π 4 centimeters 5 42π The surface area of the cylinder is about 132 square centimeters. r 5 3; h 5 4 L 5 2π(3)(4) 5 24π The lateral surface area of the cylinder is about 75 square inches. 10. Calculate the surface area of a piece of rebar that has a diameter of 1.3 centimeters and a height of 75 centimeters. Use 3.14 for p and round your answer to the nearest tenth. S 5 2π( ) 1 2π(0.65)(75) π π π The surface area of the piece of rebar is approximately square centimeters. Be prepared to share your solutions and methods. 946 Chapter 17 Volume

16 Follow Up Assignment Use the Assignment for Lesson 17.2 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson 17.2 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter 17. Check for Students Understanding Consider a cylindrical fish tank. The height of the tank is 30 inches. The length of the diameter of a base of the tank is 27.5 inches. One U.S. gallon is equal to approximately 231 cubic inches. The outside of the entire tank is coated with a protective material. Calculate the amount of protective material that covers the tank. S 5 pr 2 1 2prh 5 p( ) 1 2p(13.75)(30) square inches About square inches of protective material covers the tank Surface Area of a Cylinder 946A

17 946B Chapter 17 Volume

18 Practice Makes Perfect Volume and Surface Area Problems Learning Goals In this lesson, you will: Use the volume of a cylinder formula to solve problems. Use the surface area of a cylinder to solve problems. Use the volume of a cone formula to solve problems. Use the volume of a sphere formula to solve problems. Essential Ideas The volume formulas for a cylinder, cone, and sphere are used to solve problems within different contexts. The formula for the volume of a cylinder is V 5 pr 2 h, where V is the volume of the cylinder, r is the length of the radius of the base of the cylinder, and h is the height of the cylinder. The formula for the surface area of a cylinder is S 5 2pr 2 1 2prh, where S is the surface area of the cylinder, r is the length of the radius of a base of the cylinder, and h is the height of the cylinder. The formula for the volume of a cone is V r2 h, where V is the volume of the cone, r is the length of the radius of the base of the cone, and h is the height of the cone. The formula for the volume of a sphere is 4 3 pr3. Texas Essential Knowledge and Skills for Mathematics Grade 8 (6) Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. The student is expected to: (A) describe the volume formula V 5 Bh of a cylinder in terms of its base area and its height (7) Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to: (A) solve problems involving the volume of cylinders, cones, and spheres (B) use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders 17.5 Volume and Surface Area Problems 969A

19 Overview Students review all volume, surface area, and lateral area formulas they have learned up to this point. Students determine the surface areas of two different cereal boxes, the volume of grain needed to fill a silo, the amount of material needed to coat a prism, the volume of a cone and a melted scoop of yogurt, and the volume of a cylindrical and cone shaped popcorn container. They also compare the surface area and lateral area of a piece of rebar. 969B Chapter 17 Volume

20 Warm Up You have been asked to supply the ice cream for a birthday party. Assume each person attending the party will eat one ice cream cone. The local grocery store sells ice cream in rectangular half-gallon containers. Each container is 6.75 inches in length, 5 inches in height, and 3.5 inches in width. Your plan is to put two scoops of ice cream in each cone. Each scoop is a sphere with a radius of 1 inch. 1. What is the volume of one scoop of ice cream? V p (1)3 V p (1) V < in 3 The volume of one scoop of ice cream is approximately 4.19 in What is the volume of two scoops of ice cream? V < 2(4.19) < 8.38 in 3 The volume of two scoops of ice cream is approximately 8.38 in What is the volume of ten scoops of ice cream? V < 10(4.19) < 41.9 in 3 The volume of ten scoops of ice cream is approximately 41.9 in What is the volume of one half-gallon of ice cream? V 5 (6.75)(5)(3.5) in 3 The volume of one half-gallon of ice cream is in If 30 people attended the birthday party, how many half-gallons of ice cream should you buy? V < (8.38)(30) < in 3 (amount of ice cream needed for 2 scoops per person) V 5 ( )(2) V 5 ( )(3) I will need to buy three one half-gallons of ice cream to have enough for 30 people Volume and Surface Area Problems 969C

21 969D Chapter 17 Volume

22 Practice Makes Perfect Volume and Surface Area Problems Learning Goals In this lesson, you will: Use the volume of a cylinder formula to solve problems. Use the surface area of a cylinder to solve problems. Use the volume of a cone formula to solve problems. Use the volume of a sphere formula to solve problems. How can you tell when a word problem is asking you about volume and not about something else like lateral area or surface area? What about clue words? Take the words fill and cover, for example. Which one might refer better to surface area, and which one might refer better to volume? What other strategies can you use to tell the difference? Can you identify strategies you can use to figure out when a problem is asking you about volume? What if a problem is asking you about surface area? How about lateral area? 17.5 Volume and Surface Area Problems 969

23 Problem 1 Students write the formulas for the volume, lateral area, and surface area for all of the solids they have learned up to this point. Grouping Have students complete Questions 1 and 2 with a partner. Then share the results as a class. Problem 1 Formula Review By now, you have determined some formulas for prisms, pyramids, cylinders, cones, and spheres. 1. Complete the formula for the volume, surface area, and lateral area for each solid in the table. Use V for volume, L for lateral area, S for surface area, B for area of a base, P for perimeter of a base, h for height, r for radius, l for slant height, and π for pi. Solid Model Volume Lateral Area Surface Area Rectangular Prism V 5 Bh L 5 Ph S 5 Ph 1 2B Share Phase, Question 1 Can you determine similarities among the formulas? What are some major differences among the formulas? Which formulas do you have the most difficulty remembering? Triangular Prism Rectangular Pyramid Triangular Pyramid V 5 Bh L 5 Ph S 5 Ph 1 2B V Bh L Pl S Pl 1 B V Bh L Pl S Pl 1 B Cylinder V 5 πr 2 h L 5 2πrh S 5 2πrh 1 2πr 2 2. State the volume formula for each solid. Solid Model Volume Cone V πr2 h Sphere V πr3 970 Chapter 17 Volume

24 Problem 2 Students use the formula for the surface area of a rectangular prism to determine which cereal box uses less cardboard. Grouping Have students complete the problem with a partner. Then share the results as a class. Problem 2 Cereal Boxes A cereal manufacturer is designing a new cereal box to try and cut costs. They felt their old design used too much cardboard to build. They believe their new design, being taller and thinner will use less cardboard. Determine which cereal box design will use less cardboard to build. 74 cm 53 cm Share Phase, Cereal Boxes Problem How did you determine which formula to use? What information in this problem is helpful when determining which design uses less cardboard? 9 cm 36.5 cm Old Box Design Surface Area of Old Box Design: S 5 Ph 1 2B 5 ( )(53) 1 2(36.5? 9) 5 (91)(53) 1 2(328.5) cm 20.5 cm New Box Design Surface Area of New Box Design: S 5 Ph 1 2B 5 ( )(74) 1 2(20.5? 6.5) 5 (54)(74) 1 2(133.25) The surface area of the old box design is 5480 square centimeters and the surface area of the new box design is square centimeters. So, the new design will use less cardboard to build because it has the lesser surface area Volume and Surface Area Problems 971

25 Problem 3 A silo is composed of a hemisphere and a cylinder. Students are given the radius of the hemisphere/base of the cylinder and the height of the cylinder. Using this information they will calculate the volume of silo. The length, width, and height of a rectangular container attached to a truck that hauls the grain is given and students determine the number of truckloads of grain needed to fill the silo. Grouping Have students complete the problem with a partner. Then share the results as a class. Share Phase, Silo Problem Have you even seen a silo? Where? What do you suppose was stored in the silo? Aside from storing grain, what else are silo s used for? To determine the volume of the hemisphere, what volume formula is used? To determine the volume of the cylinder, what volume formula is used? What information in the problem is helpful when determining the volume of the hemisphere? What information in the problem is helpful when determining the volume of the cylinder? Problem 3 The Silo A silo is used to store grain that farm animals eat during the winter months. The top of the silo is a hemisphere with a radius of 8 ft. The cylindrical body of the silo shares the same radius as the hemisphere and has a height of 40 ft. The truck hauling grain to the silo has a rectangular container attached to the back that is 8 feet in length, 5 feet in width, and 4 feet in height. Determine the number of truckloads of grain required to fill an empty silo. V 5 ( 4 3 ) πr3 V 5 ( 4 3 ) π(8)3 V V V 5 πr 2 h V 5 π(8) 2 (40) V The volume of grain needed to fill the silo is approximately cubic feet The volume of a truckload is 160 cubic feet It would require 57 truckloads to fill the silo. Use 3.14 for pi. What unit of measure is used when describing the volume of the hemisphere and cylinder? How did you determine the amount of grain in one truckload? 972 Chapter 17 Volume

26 Problem 4 Students use the formula for the lateral surface area of a triangular prism to determine the amount of material used to coat an optical prism. Grouping Have students complete the problem with a partner. Then share the results as a class. Share Phase, Optical Prism Problem Have you ever seen an optical prism? How did you know which formula to use to solve this problem? What information in this problem is helpful when determining the amount of material used to coat the prism? Problem 4 Optical Prism Prisms are used in optical instruments like binoculars, telescopes, and microscopes. Optical prisms are used to bend light beams, reflect light beams, or break up light beams into separate colors. Optical prisms like the one shown reflect light beams. The arrows show the path of a light beam as it travels through the prism. 10 millimeters 6 millimeters 5 millimeters 8 millimeters The faces of optical prisms can be coated with a material so that they are not reflective and do not interfere with the path of a light beam. How much material is needed to coat the prism shown so that it is not reflective? L 5 Ph 5 ( )(5) 5 24(5) The lateral surface area of the prism is 120 square millimeters. So, 120 square millimeters of material would be needed to coat the prism so that it is not reflective Volume and Surface Area Problems 973

27 Problem 5 One circular scoop of yogurt is placed on the top of a cone. Students are given the height of the cone and diameter of the circular scoop of yogurt/circular base of the cone. Students use the volume formulas to determine if when the frozen scoop of yogurt melts, it will overflow the cone. Have English Language Learners practice reasoning and speaking with increasing accuracy by asking volunteers to answer questions about Problem 4, such as: What would you say if someone wrote V p(6)2 (12) for the volume of the yogurt cone? How you would solve the problem? Describe it in words. Problem 5 Frozen Yogurt Cone The frozen yogurt cone is 12 cm in height and has a diameter of 6 cm. A scoop of frozen yogurt is placed on the wide end of the cone. The scoop is a sphere with a diameter of 6 cm. If the scoop of frozen yogurt melts into the cone, will the cone overflow? Explain your reasoning. V 5 ( 1 3 ) πr2 h V 5 ( 1 3 ) π(3)2 (12) V The volume of the cone is approximately cubic centimeters. V 5 ( 4 3 ) πr3 V 5 ( 4 3 ) π(3)3 V The volume of the sphere is approximately cubic centimeters. Since the volume of the sphere and the volume of the cone are equal, it will not overflow. Grouping Have students complete the problem with a partner. Then share the results as a class. Share Phase, Frozen Yogurt Cone Problem To determine the volume of the scoop of frozen yogurt, what volume formula is used? To determine the volume of the cone, what volume formula is used? What information in the problem is helpful when determining the volume of the scoop of frozen yogurt? What information in the problem is helpful when determining the volume of the cone? What unit of measure is used when describing the volume of the scoop of yogurt and cone? How did you determine if there was an overflow? 974 Chapter 17 Volume

28 Problem 6 A cylindrical and conical container of popcorn holding the same amount of popcorn is shown. Students are given the radius of the circular base of the cylinder and the height of the cylinder. Using this information, they will determine volume of the cylindrical container, and the radius and height of the conical container. Students then determine the height of the cone if the radius of conical container is the same as the radius of the cylindrical container. Problem 6 Containers of Popcorn Consider the cylindrical tub of popcorn shown. 3 in. POPCORN 6 in. Consider the conical container of popcorn shown. POPCORN Grouping Have students complete Questions 1 through 3 with a partner. Then share the results as a class. Assume the conical container and the cylindrical tub of popcorn are full and they hold the same amount of popcorn. Share Phase, Questions 1 through 3 To determine the volume of the cylindrical container, what volume formula is used? What information in the problem is helpful when determining the volume of the cylindrical container? What unit of measure is used when describing the volume of the cylindrical container? To determine the radius and height of the conical container, what volume formula is used? What information in the problem is helpful when determining the radius and height of the conical container? 1. Calculate the volume of the cylindrical tub. V 5 πr 2 h V 5 π(3) 2 (6) V The volume of the cylindrical tub is approximately cubic inches. What unit of measure is used when describing the radius and height of the conical container? Is there more than one correct answer? Explain Volume and Surface Area Problems 975

29 2. If the height of the conical container is 10.1 inches, what is the radius of the cone? V 5 ( 1 3 ) πr2 h ( 1 3 ) πr (169.56) 5 3 ( ( 1 3 ) πr ) πr πr πr πr2 π 5 π r r 4 r If the height is approximately 10.1 inches, the radius is approximately 4 inches. 3. If the radius of the conical container is the same as the radius of the cylindrical tub, what is the height of the cone? V 5 ( 1 3 ) πr2 h ( 1 3 ) π32 h 3(169.56) 5 3 ( ( 1 3 ) π32 h ) π3 2 h π9h πh π 5 πh π 18 h If the radius of the cone is 3 inches, the height would have to be approximately 18 inches. 976 Chapter 17 Volume

30 Problem 7 Students use the formulas for the lateral surface area and total surface area of a cylinder. Then, students compare the lateral surface area to the total surface area. Grouping Have students complete the problem with a partner. Then share the results as a class. Share Phase, Questions 1 through 3 Why do you think rebar makes concrete stronger? How did you know which formulas to use to solve this problem? Problem 7 Rebar When highway construction workers prepare a roadway for a concrete surface, they lay out steel reinforcing bars, also known as rebar. The rebar makes the concrete surface stronger. A simple model of a piece of rebar is shown. 13 millimeters A shipment of 200 pieces of rebar has arrived for a construction crew. Each piece of rebar is 2 meters long and has the same diameter as the model of the piece of rebar shown. All 200 pieces of rebar must be coated with a special sealant before the construction crew can use them. 1. What is the total amount of sealant needed to coat all of the rebar? The radius is equal to 6.5 millimeters, or meter. S 5 2πr 2 1 2πrh 2(3.14)(0.0065) 2 1 2(3.14)(0.0065)(2) The surface area of one piece of rebar is about square meter. So, a total of about (200) square meters of sealant is needed to cover all 200 pieces of the rebar Volume and Surface Area Problems 977

31 2. If the bases of each piece of rebar do not need to be coated with sealant, how much sealant is needed? L 5 2πrh 2(3.14)(0.0065)(2) The lateral area of one piece of rebar is about square meter. So, a total of about (200) square meters of sealant is needed. 3. Compare your answers to Questions 1 and 2. Are your answers close? If so, explain why. If not, explain why not. The answers are close. The formulas for the surface area and lateral area of a cylinder differ in that the areas of the bases of a cylinder are included in the surface area, but not the lateral area. Because the areas of the bases of each cylinder are so small when compared to the lateral area, the surface area and lateral area of each cylinder are almost the same. 978 Chapter 17 Volume

32 Problem 8 Students use what they know about volume and surface area formulas to solve more complicated problems. Grouping Have students complete Questions 1 through 3 with a partner. Then share the results as a class. Share Phase, Questions 1 through 3 How can you use algebra to help you answer Question 1? What is the decimal equivalent of 25%? Can you draw a sketch to support your answer to Question 2? What formulas can help you to answer Question 3? Problem 8 Problems That Make You Say, Hmmm A frozen yogurt shop advertises that their frozen yogurt is now sold in cones which hold 25% more frozen yogurt than the old cones. The old cone has a radius of 3.75 centimeters and a height of 11 centimeters. The radius of the new cone is the same as the radius of the old cone. What is the height of the new cone? Let V o represent the volume of the old cone, and let V n represent the volume of the new cone. I know that the new cone holds 25% more frozen yogurt than the old cone. So, V n V o. Let h o represent the height of the old cone, and let h n represent the height of the new cone. Let r represent the radius of either cone. So, V o πr2 h o π(3.75)2 (11) π and V n πr2 h n π(3.75)2 h n πh n. V n V o πh n ( π) h n ( ) h n h n The height of the new cone is centimeters Volume and Surface Area Problems 979

33 2. Is it possible for two cylinders to have equivalent lateral areas and different surface areas? If so, give an example. If not, explain why not. Yes. It is possible. In order for the lateral areas to be equivalent, the product of the radius and height of one cylinder must be equivalent to the product of the radius and height of the other cylinder. In order for the surface areas to be different, the radii of the cylinders must be different and/or the heights of the cylinders must be different. For example, suppose that Cylinder A has a radius of 4 units and a height of 3 units. The product of the radius and height is 12. Suppose that Cylinder B has a radius of 2 units and a height of 6 units. The product of the radius and height is also 12. So, the products are the same and the radii are not equal and the heights are not equal. Cylinder A: Cylinder B: L 5 2πrh L 5 2πrh 5 2π(4)(3) 5 2π(2)(6) 5 24π 5 24π S 5 2πrh 1 2πr 2 S 5 2πrh 1 2πr π 1 32π 5 24π 1 8π 5 56π 5 32π So, the lateral areas of the cylinders are equivalent and the surface areas of the cylinders are different. 980 Chapter 17 Volume

34 3. Two cylinders have the same height and the same lateral area. Bethany concludes that the two cylinders must have the same volume. Ella says that the cylinders may have the same volume, but their volumes could also be different. Who is correct? Bethany is correct. The formula for the lateral area of a cylinder is L 5 2πrh. If the heights of the cylinders are the same and the lateral areas of the cylinders are the same, then the radii of the cylinders must be the same. The formula for the volume of a cylinder is V 5 2πr 2 h. So, if the heights of the cylinders are the same and the radii of the cylinders are the same, then the volumes of the cylinders must be the same. Be prepared to share you solutions and methods Volume and Surface Area Problems 981

35 Follow Up Assignment Use the Assignment for Lesson 17.5 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson 17.5 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter 17. Check for Students Understanding Your neighbor, a contractor, has agreed to help you repave your driveway. He is requesting that you purchase the premixed cement at a local lumber yard. Premixed cement is sold by the cubic foot. Calculate the number of cubic yards needed to pave a rectangular driveway that is 55 feet in length, 15 feet in width, and 6 inches in depth or thickness. V 5 (length)(width)(height) V 5 (55)(15)(0.5) V The driveway would require approximately cubic feet of cement. 982 Chapter 17 Volume

36 Chapter 17 Summary Key Terms cylinder (17.1) right circular cylinder (17.1) radius of a cylinder (17.1) height of a cylinder (17.1) circumference (17.1) pi (17.1) surface area of a cylinder (17.2) cone (17.3) height of a cone (17.3) sphere (17.4) center of a sphere (17.4) radius of a sphere (17.4) diameter of a sphere (17.4) antipodes (17.4) great circle (17.4) hemisphere (17.4) Calculating the Volume of Right Circular Cylinders A cylinder is a three-dimensional object with two parallel, congruent, circular bases. A right circular cylinder is a cylinder in which the bases are aligned one directly above the other. The two circular portions of the cylinder represent the bases. The area of the base can be determined by using the formula A 5 πr 2, where A represents the area of the circle, and r represents the radius of the circle. The volume of a cylinder can be calculated by multiplying the area of the cylinder s base times the height of the cylinder. Example 4 cm 3 cm The diameter of the base is 4 cm. Therefore, the radius of the base is 2 cm. The area of the base is π (4) cm 2. The volume of the cylinder is cm 3. Chapter 17 Summary 983

37 Calculating Surface Area of Cylinders A cylinder is a solid formed by two congruent circles situated on parallel planes. The surface area of a cylinder is the total area of the outside surfaces of the cylinder. To calculate the surface area of a cylinder, use the formula S 5 2pr 2 1 2prh, where S is the surface area of the cylinder, r is the radius of the cylinder, and h is the height of the cylinder. Example 6.8 cm 11.2 cm S 5 2pr 2 1 2prh 5 2p(6.8 2 ) 1 2p(6.8)(11.2) 2(3.14)(46.24) 1 2(3.14)(6.8)(11.2) The surface area of the cylinder is approximately square centimeters. Calculating the Volume of Cones A cone is a three-dimensional figure with a circular or elliptical base and one vertex. All of the cones associated with this chapter have a circular base and a vertex that is located directly above the center point of the base of the cone. To calculate the volume of a cone, use the formula V 5 1 Bh, where V represents the volume of the cone, B represents the area 3 of the base, and h represents the height. Example 4 cm 9 cm The area of the base is π 9 2 < 3.14(81) < cm 2. The volume of the cone is 1 3 ( ) cm Chapter 17 Volume

38 Calculating the Volume of Spheres A sphere is the set of all points in three dimensions that are equidistant from a center point. To calculate the volume of a sphere, use the formula V πr3, where V represents the volume of the sphere, and r represents the radius of the sphere. Example The radius of a tennis ball is 3.35 cm. Use the radius to find the volume of the tennis ball. V π(3.353 ) < 4 3 (3.14)(37.60) < The volume of a tennis ball is about cm 3. Using Surface Area Formulas to Solve Problems The formulas for surface area of many three-dimensional shapes can be used to solve real-life scenarios. Example Penelope places a gift in a gift box and wraps it for her aunt. How much wrapping paper does she use to cover the box? 14 in. 7 in. 30 in. Surface Area of Rectangular Prism: S 5 Ph 1 2B 5 ( )(14) 1 2(7 30) 5 74(14) 1 2(210) The surface area of the rectangular prism is 1456 square inches. So, Penelope needs 1456 square inches of wrapping paper. Chapter 17 Summary 985

39 Using Surface Area Formulas to Solve Problems The formulas for surface area of many three-dimensional shapes can be used to solve real-life scenarios. Example Amit is making blocks for his nephew. He carves the blocks out of wood and then paints them orange. How much paint will he need to paint this block? 15 cm 9 cm 16 cm 12 cm Surface Area of Triangular Prism: S 5 Ph 1 2B 5 ( )(16) 1 2 ( ) 5 36(16) 1 2(54) The surface area of the triangular prism is 684 square centimeters. So, Amit needs 684 square centimeters of orange paint. 986 Chapter 17 Volume

40 Using Volume Formulas to Solve Problems The formulas for volume of many three-dimensional shapes can be used to solve real-life scenarios. Example The volume of The Sphere sculpture by Fritz Koenig that used to stand between the World Trade Center towers in New York City is ft 3. The formula for the volume of a sphere can be used to find the circumference of the sculpture. V πr πr3 3 3 ( ) 5 4 4( 4 3 ) πr 3 C 5 2πr πr 3 5 2π(7.5) Speaking of volume, did you know exercise can increase the volume of your brain? Another reason to get moving! π 5 πr3 π < < r 3 3 œ wwww < œw 3 r 7.5 < r So, the circumference of the sphere is approximately 47.1 ft. Chapter 17 Summary 987

41 988 Chapter 17 Volume

42 Surface Area Go Around Activity Quick Check

43 Item 1 Find TOTAL surface area and LATERAL surface area.

44 Item 2 Find TOTAL surface area and LATERAL surface area.

45 Item 3 Find TOTAL surface area and LATERAL surface area.

46 Item 4 Find TOTAL surface area and LATERAL surface area.

47 Item 5 Find TOTAL surface area and LATERAL surface area.

48 Item 6 Find TOTAL surface area and LATERAL surface area.

49 Item 7 Find TOTAL surface area and LATERAL surface area.

50 Item 8 Find TOTAL surface area and LATERAL surface area.

51 Name: Date: Pd: Surface Area Go-Around Be sure to show your work, even if using a calculator. Item # Lateral Surface Area Total Surface Area 1 2 3

52

53 Scratching the Surface Surface Area of a Cylinder Learning Goals In this lesson, you will: Determine the surface area of cylinders. Key Terms surface area of a cylinder Archimedes of Syracuse, Sicily (c BC), was an ancient Greek mathematician, physicist, and engineer. Among other things, Archimedes discovered formulas involving cylinders. Archimedes considered his greatest achievement to be his discovery of the relationship between a cylinder and another shape which you will learn about later in this chapter. Archimedes must have really liked cylinders, because he even asked for a sculpted cylinder to be placed on his tomb! In this lesson, you will determine a formula for the surface area of a cylinder. Think about the pieces that make up a net of a cylinder. How do you think the formula for the surface area of a cylinder will be different than the formula for the volume of a cylinder? 17.2 Surface Area of a Cylinder 941

54 Problem 1 Poster Board 36 in. 24 in. A standard piece of poster board is 24 inches by 36 inches. A cylindrical shape can be made by taping two ends of the poster board together. 1. Explain which way the poster board should be taped to result in the cylinder having the greatest volume. Justify your conclusion. 942 Chapter 17 Volume

55 Problem 2 Surface Area of a Cylinder Recall that the surface area of a solid three-dimensional object is the total area of the outside surfaces (faces and bases) of the solid. Surface area is described using square units of measure. It follows that the surface area of a cylinder is the total area of the outside surfaces of the cylinder. 1. Clarissa, Xavier, and Umi each drew a different net, as shown. Clarissa s Net Xavier s Net Umi s Net Whose net represents the net of a cylinder? Explain your reasoning Surface Area of a Cylinder 943

56 2. On another piece of paper, sketch a larger version of one of the nets from the previous question. Then cut out the net and tape the longer sides together to form a cylinder. a. What shapes were used to form your model of a cylinder? b. Use your model to help you explain how you would calculate the surface area of a cylinder. 3. Remove the tape from your model of a cylinder. a. Calculate the length of the rectangle. Use a ruler to calculate your measurements. b. How does the width of the rectangle relate to the circle? c. Calculate the width of the rectangle. d. Calculate the area of the rectangle. 944 Chapter 17 Volume

57 4. Calculate the area of the bases of your model of a cylinder. Use 3.14 for π. Recall that the lateral surface area of a solid three-dimensional object is the sum of the surface areas of all its faces excluding the bases of the solid. 5. Calculate the lateral surface area of your model of a cylinder. Use 3.14 for π. 6. Write a formula for the lateral surface area of a cylinder. Use L for the lateral surface area, r for the radius, h for the height, and π for pi. 7. Calculate the surface area of your model of a cylinder. 8. Write a formula for the surface area of a cylinder. Use S for the surface area, r for the radius, h for the height, and π for pi Surface Area of a Cylinder 945

58 9. Calculate the lateral surface area and the total surface area of each cylinder. Use 3.14 for π. Round your answer to the nearest whole unit of measure if necessary. a. 6 inches 20 inches b. 6 centimeters 4 centimeters 10. Calculate the surface area of a piece of rebar that has a diameter of 1.3 centimeters and a height of 75 centimeters. Use 3.14 for p and round your answer to the nearest tenth. Be prepared to share your solutions and methods. 946 Chapter 17 Volume

59 Practice Makes Perfect Volume and Surface Area Problems Learning Goals In this lesson, you will: Use the volume of a cylinder formula to solve problems. Use the surface area of a cylinder to solve problems. Use the volume of a cone formula to solve problems. Use the volume of a sphere formula to solve problems. How can you tell when a word problem is asking you about volume and not about something else like lateral area or surface area? What about clue words? Take the words fill and cover, for example. Which one might refer better to surface area, and which one might refer better to volume? What other strategies can you use to tell the difference? Can you identify strategies you can use to figure out when a problem is asking you about volume? What if a problem is asking you about surface area? How about lateral area? 17.5 Volume and Surface Area Problems 969

60 Problem 1 Formula Review By now, you have determined some formulas for prisms, pyramids, cylinders, cones, and spheres. 1. Complete the formula for the volume, surface area, and lateral area for each solid in the table. Use V for volume, L for lateral area, S for surface area, B for area of a base, P for perimeter of a base, h for height, r for radius, l for slant height, and π for pi. Solid Model Volume Lateral Area Surface Area Rectangular Prism Triangular Prism Rectangular Pyramid Triangular Pyramid Cylinder 2. State the volume formula for each solid. Solid Model Volume Cone Sphere 970 Chapter 17 Volume

61 Problem 2 Cereal Boxes A cereal manufacturer is designing a new cereal box to try and cut costs. They felt their old design used too much cardboard to build. They believe their new design, being taller and thinner will use less cardboard. Determine which cereal box design will use less cardboard to build. 53 cm 74 cm 9 cm 36.5 cm Old Box Design 6.5 cm 20.5 cm New Box Design 17.5 Volume and Surface Area Problems 971

62 Problem 3 The Silo A silo is used to store grain that farm animals eat during the winter months. The top of the silo is a hemisphere with a radius of 8 ft. The cylindrical body of the silo shares the same radius as the hemisphere and has a height of 40 ft. The truck hauling grain to the silo has a rectangular container attached to the back that is 8 feet in length, 5 feet in width, and 4 feet in height. Determine the number of truckloads of grain required to fill an empty silo. Use 3.14 for pi. 972 Chapter 17 Volume

63 Problem 4 Optical Prism Prisms are used in optical instruments like binoculars, telescopes, and microscopes. Optical prisms are used to bend light beams, reflect light beams, or break up light beams into separate colors. Optical prisms like the one shown reflect light beams. The arrows show the path of a light beam as it travels through the prism. 10 millimeters 6 millimeters 8 millimeters 5 millimeters The faces of optical prisms can be coated with a material so that they are not reflective and do not interfere with the path of a light beam. How much material is needed to coat the prism shown so that it is not reflective? 17.5 Volume and Surface Area Problems 973

64 Problem 5 Frozen Yogurt Cone The frozen yogurt cone is 12 cm in height and has a diameter of 6 cm. A scoop of frozen yogurt is placed on the wide end of the cone. The scoop is a sphere with a diameter of 6 cm. If the scoop of frozen yogurt melts into the cone, will the cone overflow? Explain your reasoning. 974 Chapter 17 Volume

65 Problem 6 Containers of Popcorn Consider the cylindrical tub of popcorn shown. Consider the conical container of popcorn shown. 3 in. POPCORN 6 in. POPCORN Assume the conical container and the cylindrical tub of popcorn are full and they hold the same amount of popcorn. 1. Calculate the volume of the cylindrical tub Volume and Surface Area Problems 975

66 2. If the height of the conical container is 10.1 inches, what is the radius of the cone? 3. If the radius of the conical container is the same as the radius of the cylindrical tub, what is the height of the cone? 976 Chapter 17 Volume

67 Problem 7 Rebar When highway construction workers prepare a roadway for a concrete surface, they lay out steel reinforcing bars, also known as rebar. The rebar makes the concrete surface stronger. A simple model of a piece of rebar is shown. 13 millimeters A shipment of 200 pieces of rebar has arrived for a construction crew. Each piece of rebar is 2 meters long and has the same diameter as the model of the piece of rebar shown. All 200 pieces of rebar must be coated with a special sealant before the construction crew can use them. 1. What is the total amount of sealant needed to coat all of the rebar? 17.5 Volume and Surface Area Problems 977

68 2. If the bases of each piece of rebar do not need to be coated with sealant, how much sealant is needed? 3. Compare your answers to Questions 1 and 2. Are your answers close? If so, explain why. If not, explain why not. 978 Chapter 17 Volume

69 Problem 8 Problems That Make You Say, Hmmm A frozen yogurt shop advertises that their frozen yogurt is now sold in cones which hold 25% more frozen yogurt than the old cones. The old cone has a radius of 3.75 centimeters and a height of 11 centimeters. The radius of the new cone is the same as the radius of the old cone. What is the height of the new cone? 17.5 Volume and Surface Area Problems 979

70 2. Is it possible for two cylinders to have equivalent lateral areas and different surface areas? If so, give an example. If not, explain why not. 980 Chapter 17 Volume

71 3. Two cylinders have the same height and the same lateral area. Bethany concludes that the two cylinders must have the same volume. Ella says that the cylinders may have the same volume, but their volumes could also be different. Who is correct? Be prepared to share you solutions and methods Volume and Surface Area Problems 981

72 982 Chapter 17 Volume

73 Lesson 17.2 Assignment Name Date Scratching the Surface Surface Area of a Cylinder A construction crew is building a new tunnel through a mountain to connect two neighboring towns. 1. Prior to construction, the crew sets up parking barrels to divert traffic away from the construction site. The crew must paint the parking barrels orange. In order to determine the amount of paint needed, calculate the surface area of the barrel. Use 3.14 for pi. 24 in. 54 in. Chapter 17 Assignments 293

74 Lesson 17.2 Assignment page 2 2. The tunnel through the mountain is in the shape of half of a cylinder. The entrance to the tunnel is 40 feet wide, as shown. The tunnel is 800 feet long. 40 ft The outside of the tunnel must be coated in cement. In order to determine the amount of cement needed, calculate the surface area of the tunnel. Use 3.14 for pi. 294 Chapter 17 Assignments

75 Lesson 17.5 Assignment Name Date Practice Makes Perfect Volume and Surface Area Problems Veronica owns a kid s party planning business. She helps decide the theme of the party, the games, food, and location of the party for busy parents. Her next party has a summer time theme. 1. One of Veronica s specialties is making homemade party-themed piñatas. She makes the piñatas and stuffs them full of candy. For the summer party she is making a large sphere-shaped piñata decorated as the sun. The piñata is going to be 2 feet in diameter. a. In order to know how much candy she will need to fill this large piñata, she needs to know the volume of the piñata. Calculate the volume of this piñata. Use 3.14 for pi. b. A regular size piñata has a volume of about 2 cubic feet, and it holds two 2-lb bags of mixed candy. How many bags will the large, sun-shaped piñata hold? Chapter 17 Assignments 301

76 Lesson 17.5 Assignment page 2 2. Veronica is serving snow cones at the party. The snow cone cups are in the shape of a cone. These cups have a height of 5 inches and a radius of 1.5 inches. Veronica wants to know how much ice can fit into each cup. Calculate the volume of the cup. Use 3.14 for pi. 3. For party favors, Veronica is handing out finger puzzles. She wants to paint the outside of each puzzle a fun summer color. In order to determine the amount of paint needed, calculate the surface area of one puzzle. Use 3.14 for pi. 1 in. 5 in. 302 Chapter 17 Assignments

77 Lesson 17.2 Skills Practice Name Date Scratching the Surface Surface Area of a Cylinder Vocabulary Define the term in your own words and identify the formula for solving. 1. surface area of a cylinder: The surface area of a cylinder is the total area of the outside surfaces of the cylinder. The formula to determine the surface area of a cylinder is SA 5 2p rh 1 2p r 2. Problem Set Suppose that each cylinder is unfolded to form a net. Calculate the area of the rectangle in each net. Use 3.14 for p. Round decimals to the nearest tenth if necessary in m 7 in. 8 m A < 2(3.14)(7)(7) A < i n 2 A < 2(3.14)(6)(8) A < m ft 16 ft in. 5 in. A < 2(3.14)(8)(2) A < f t 2 A < 2(3.14)(3.2)(5) A < i n. 2 Chapter 17 Skills Practice 1025

78 Lesson 17.2 Skills Practice page yd 3 m 10 yd 2.1 m A < 2(3.14)(1.5)(2.1) A < 19.8 m 2 A < 2(3.14)(3.5)(10) A < y d 2 Calculate the surface area of each cylinder. Use 3.14 for p. Round decimals to the nearest tenth if necessary in m 7 in. 8 m SA 5 2p rh 1 2p r 2 SA 5 2p (7)(7) 1 2p (7 ) 2 SA 5 196p SA i n. 2 SA 5 2p r h 1 2p r 2 SA 5 2p (6)(8) 1 2p (6 ) 2 SA 5 168p SA m ft in. 5 in. 6 ft SA 5 2p r h 1 2p r 2 SA 5 2p ( 6 2 ) (2) 1 2p ( 6 SA 5 30p SA 94.2 f t 2 2 ) 2 SA 5 2p r h 1 2p r 2 SA 5 2p ( 3.2 SA p SA 66.3 i n. 2 2 ) (5) 1 2p ( ) Chapter 17 Skills Practice

79 Lesson 17.2 Skills Practice page 3 Name Date cm ft 7.4 cm 4 ft SA 5 2p r h 1 2p r 2 SA 5 2p r h 1 2p r 2 SA 5 2p (4)(7.4) 1 2p (4 ) 2 SA p SA 5 2p (10)(4) 1 2p (10 ) 2 SA 5 280p SA f t 2 SA c m ft yd 7.5 ft 32 yd SA 5 2p r h 1 2p r 2 SA 5 2p ( 5 2 ) (7.5) 1 2p ( 5 SA 5 50p SA 157 f t 2 2 ) 2 SA 5 2p r h 1 2p r 2 SA 5 2p ( 15 SA p 2 ) (32) 1 2p ( 15 2 ) 2 SA y d 2 Chapter 17 Skills Practice 1027

80 1028 Chapter 17 Skills Practice

81 1046 Chapter 17 Skills Practice

82 Lesson 17.5 Skills Practice Name Date Practice Makes Perfect Volume and Surface Area Problems Problem Set Identify the three-dimensional solid and the measurement determined by each formula Bh 2. Ph 1 2B 3 1 The formula Bh can be used to The formula Ph 1 2B can be used 3 determine the volume of a rectangular to determine the surface area of a or triangular pyramid. rectangular or triangular prism. 3. 2p r h p r 3 4 The formula 2p rh can be used to The formula 3 p r 3 can be used to determine the lateral area of determine the volume of a a cylinder. sphere p r 2 h Pl 1 B 1 The formula 3 p r 2 h can be used to The formula Pl 1 B can be used to 2 determine the volume of a cone. determine the surface area of a rectangular or triangular pyramid. Chapter 17 Skills Practice 1047

83 Lesson 17.5 Skills Practice page 2 Use the formulas for the volume of a cone, a sphere, and a cylinder to solve each problem. Use 3.14 for pi. 7. Which paper cup can hold more and by how much: the cone or the cylinder? V 5 πr 1.5 in. 2 in. 2 h V πr2 h 3 in. 4.5 in. 5 π (1.5) 2 (3) π π(2)2 (4.5) 5 6π < in. 3 < in The cylindrical cup holds about cubic inches more than the conical cup Chapter 17 Skills Practice

84 Lesson 17.5 Skills Practice page 3 Name Date 8. Calculate the total volume of the Erlenmeyer flask. 4 cm 10 cm V πr2 h V πr2 h π(10)2 (30) π π(2)2 (10) < 26.67π < 3140 < cm cm The volume of the flask is approximately cubic centimeters. Chapter 17 Skills Practice 1049

85 Lesson 17.5 Skills Practice page 4 9. The drinking glass is not a cylinder, but is actually part of a cone. Determine the volume of the glass. 4.5 cm V πr2 h V πr2 h π(4.5)2 (42) π(3.5)2 (26) 16 cm 3.5 cm 26 cm π < < π < The volume of the glass is approximately cubic centimeters Chapter 17 Skills Practice

86 Lesson 17.5 Skills Practice page 5 Name Date 10. A tennis ball company is designing a new can to hold 3 tennis balls. They want to waste as little space as possible. How much space does each can waste? Which can design should they choose? 2.75 in. 5.5 in. 3V 5 4πr 3 5 4π(1.25) in. < in in. 5.5 in. V 5 πr 2 h V πr2 h 5 π(1.375) 2 (7.75) π(2.75)2 (5.5) < 14.65π < 13.86π < 46 in. 3 < 43.5 in. 3 The company should choose the conical design because it only wastes , or 19 cubic inches, while the cylindrical can wastes , or 21.5 cubic inches. Chapter 17 Skills Practice 1051

87 Lesson 17.5 Skills Practice page A candle company makes pillar candles, spherical candles, and conical candles. They have an order for 3 pillar, 2 spherical, and 1 conical candle. Wax is sold in large rectangular blocks. What are the possible dimensions for a wax block that could be used to fill this order? 1.5 in. 5 in. 2 in. 4 in. 2.5 in. V 5 πr 2 h V πr3 V πr2 h 5 π(1.5) 2 (5) π(2) π(2.5)2 (4) < 35.3 < 33.5 < (35.3) 1 2(33.5) 1 (26.2) The total volume of the candles is approximately cubic inches. The block of wax could have a volume of 200 cubic inches. So, the dimensions of the wax block that is needed could be 10 inches by 5 inches by 4 inches Chapter 17 Skills Practice

88 Lesson 17.5 Skills Practice page 7 Name Date 12. A jeweler sold a string of fifty 8-millimeter pearls. He needs to choose a box to put them in. Which box should the jeweler choose? 10 mm 25 mm B 40 mm 15 mm A V 5 πr 2 h V 5 πr 2 h V 5 50 ( 4 3 π(4)3 ) 5 π(25) 2 (15) 5 π(10) 2 (40) < 50? 268 < 29,437.5 mm 3 < 12,560 mm 3 < 13,400 mm 3 The jeweler should use box A because its volume is sufficiently large to contain the pearls and the wasted space around the pearls. Chapter 17 Skills Practice 1053

89 Lesson 17.5 Skills Practice page Chantel bought a cylindrical vase to give as a gift to a friend. She is going to wrap the vase using festive wrapping paper. Determine the amount of wrapping paper Chantel will need to wrap the vase. 3 in. 15 in. SA 5 2p r 2 1 2p r h SA 5 2p (3 ) 2 1 2p (3)(15) SA SA The surface area of the vase is square inches. So, Chantel needs i n. 2 of wrapping paper to cover the vase Chapter 17 Skills Practice

90 And That s A Wrap Investigation Activity Standards Content: 8.7B use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determining solutions for problems involving rectangular prisms, triangular prisms, and cylinders. ELPS: 3(E) share information in cooperative learning interactions 3(D) speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency Objectives Content Objective: Students will investigate surface area by using concrete 3-D shapes. Language Objective: Students will be able to explain how finding the area of all the faces on 3-D figures determines the surface area. Vocabulary Vocabulary: lateral surface area, total surface area, nets, cylinder, rectangular prism, triangular prism, perimeter, base, face Vocabulary Activity: Mix-N-Quiz 1. Each student is given a card that has a word from the Word Wall with its definition and a few facts. 2. Students stand with their card and find a partner. 3. Student A reads the definition and student B chooses a word from the Word Wall that matches the definition. 4. If student B does not choose the correct word, student A reads a fact and student B tries again. Middle School Math

91 And That s A Wrap Investigation Activity Guiding Questions How are lateral and total surface area related? What is the relationship between the sum of the area of the faces and its lateral and total surface area? What is the difference between a length unit, a square unit, and a cubic unit? How does the relationship between the dimensions of a three-dimensional figure relate to the surface area? What are strategies for finding a missing dimension when the lateral or total surface area is known? Differentiation English Language Learners: When discussing how to sort the word problems, students can use sentence stems: I know this problem is solving for lateral surface area because. I know this problem is solving for total surface area because. Extension for Learning: Activity 2 After students have completed their estimates and formulas for lateral and total surface area of each solid in square centimeters, have them estimate the surface areas in square inches. Using the one inch grid paper, have students find the ratio of square centimeters to one square inch. Students will use this ratio to convert the lateral and total surface areas of each solid from square centimeters to square inches. Ask students to describe their process for determining the ratio of square centimeters to square inches. Implementation Materials: 3-D figures for every 2-3 students (rectangular prisms, triangular prisms, cylinders) centimeter grid paper sticky dots rulers And That s A Wrap student handout STAAR 8 Reference Chart Middle School Math

92 And That s A Wrap Investigation Activity Implementation: Pass out And That's a Wrap handout to each student. Complete items 1-3 as a class. Discuss the difference between lateral surface area and total surface area. Discuss item 4 and how volume differs from surface area. Activity 1 Allow students to work in groups of 2 or 3. Distribute centimeter paper to each group. Assign each group member a different solid (rectangular prism, triangular prism or cylinder). Instruct students to place a sticky dot on each face of their designated solid(s). For example, a rectangular prism will have six sticky dots, one on each face. Students then should number the faces by writing on the sticky dots. Make sure that students number all faces. Ask students to predict the number of faces of the cylinder. Next, students will trace the net of the solid on the centimeter grid paper and number the faces on the solid. Allow students to complete Activity 1 as you circulate, support, and ask facilitating questions. o As students trace the net, they should rotate the solid so that the shared edges of the solid are between the adjacent faces of the solid on the net. o Describe how your group came up with the formulas for lateral and total surface area. Middle School Math

93 And That s A Wrap Investigation Activity Teacher note: To assist with the net of the cylinder, have students use a ruler to draw a straight line on the lateral face perpendicular to the bases. This line will serve as the starting and stopping point to mark a full rotation of the cylinder on the centimeter grid paper. As students rotate the cylinder, ask them to describe the polygon that appears. Students should say that it is a rectangle. Have students name the linear dimensions of the polygon (length and width or base and height) and describe how to find its area. Ask students to identify the base the base on the cylinder. The goal is to lead students to recognize that the circumference of the cylinder is the base of the rectangle. Therefore, in their formula, they should substitute the base with formula for circumference. Have a class discussion about their findings and answers to facilitating questions for Activity 1. o During the last part of the surface area piece, students will measure the dimensions of their solid using a ruler and calculate the lateral and total surface areas of each solid in square centimeters and square inches. Students will use the formulas they developed and substitute the measurements into the formulas. Have students compare their grid paper estimates to the measurement estimates. o Have a class discussion on the importance of precision and accuracy in measurement and the effects on costs in manufacturing. o Which measurement tool do you think is more accurate? Explain your reasoning. Extend the lesson with students completing Activity 2. Continue this topic with Carnegie Lesson 17.2 assignments and skill practice pages. Middle School Math

94 And That s A Wrap Investigation Activity Middle School Math

95 Name: Date: Period: And That's a Wrap! 1. You own a company that ships packages overnight. They want to make a profit and therefore have to keep costs of materials at a minimum. Your company wraps each package in plastic to weatherproof it and keep out moisture. Your job is to determine the least amount of plastic that can be used to cover the packages. You will have to develop a formula, as a shortcut, that will communicate to the rest of the company how to determine the amount of plastic needed for any given package. The packages come in five different shapes: Rectangular prisms Triangular prisms Cylinders 2. Describe in words the lateral surface area and write its formula for each of the five solids. Rectangular prisms Triangular prisms Cylinders

96 3. Describe in words the total surface area and write its formula for each of the three solids. Rectangular prisms Triangular prisms Cylinders 4. Which solid should your company charge a flat shipping rate on; based on what it can hold, so your company can make more profit? Rectangular prisms Triangular prisms Cylinders

97 Mathematics TEKS Refinement Student Activity Sheet 1 Tarleton State University Solid Centimeter Grid Paper Lateral Surface Area Area 1 Area 2 Area 3 Area 4 Area 5 Area 6 Formula Total Surface Area Solid Ruler Units in Centimeters Lateral Surface Area Area 1 Area 2 Area 3 Area 4 Area 5 Area 6 Formula Total Surface Area 1. Describe how your group came up with the formulas for lateral and total surface area. 2. Which measurement tool do you think is more accurate? Explain your reasoning. Measurement Grade 8 AndThat's a Wrap! Page 1

98 Mathematics TEKS Refinement Student Activity Sheet 2 Tarleton State University Solid 1-inch grid paper Lateral Surface Area Area 1 Area 2 Area 3 Area 4 Area 5 Area 6 Formula Total Surface Area Measurement Grade 8 AndThat's a Wrap! Page 2

99 Surface Area Mix N Quiz Vocab Activity Cylinder Description: solid that has two parallel (usually circular) bases connected by a curved surface. 3 Facts: Uses formula: S = 2 rh+ 2 r² A 3D figure whose net contains one rectangle and two circles The shape of a soup can Rectangular Prism Description: A solid with two congruent rectangular parallel faces, and four additional rectangular sides 3 Facts: Uses formula: S = Ph + 2B The net is six rectangles The shape of a gift box

100 Triangular Prism Description: A solid with two congruent triangular parallel faces, and three additional rectangular sides 3 Facts: Uses formula: S = Ph + 2B The base is a triangle The shape of a camping tent Total Surface Area Description: The measurement of the entire outside flat surface of a figure 3 Facts: Similar to Lateral but includes the base Take the area of each piece of a net and add it up The area of all the faces and bases added up

101 Net Description: A pattern that you can cut and fold to make a model of a solid shape. 3 Facts: Use this as one way to find total surface area Sometimes you see this with gridlines to help count units If I cut up a 3D figure and lay it out flat, it Lateral Surface Area Description: The sum of the surface areas of all its faces excluding the base of the solid. 3 Facts: Similar to total but excludes the base Take the area of each piece of a net and add it up The area of all the faces and bases added up

102 Base Description: The surface a solid object stands on, or the bottom line of a shape such as a triangle or rectangle. 3 Facts: If the top side is parallel, it s also called this A figure is named after this part of it Part of finding surface area is finding the

103 Surface Area Sort Supporting Activity Standards Content: 8.7B use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determining solutions for problems involving rectangular prisms, triangular prisms, and cylinders. ELPS: 3(E) share information in cooperative learning interactions 3(D) speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency Objectives Content Objective: Students will sort problems situations into the categories of lateral surface area or total surface area and then solve. Language Objective: Students will be able to explain how they determine whether a problem situation is asking for lateral or total surface area. Vocabulary Vocabulary: lateral surface area, total surface area, nets, cylinder, rectangular prism, triangular prism, perimeter, base, face Vocabulary Activity: Mix-N-Quiz 1. Each student is given a card that has a word from the Word Wall with its definition and a few facts. 2. Students stand with their card and find a partner. 3. Student A reads the definition and student B chooses a word from the Word Wall that matches the definition. 4. If student B does not choose the correct word, student A reads a fact and student B tries again. Middle School Math

104 Surface Area Sort Supporting Activity Guiding Questions How are lateral and total surface area related? What is the relationship between the sum of the area of the faces and its lateral and total surface area? What is the difference between a length unit, a square unit, and a cubic unit? How does the relationship between the dimensions of a three-dimensional figure relate to the surface area? What are strategies for finding a missing dimension when the lateral or total surface area is known? Differentiation English Language Learners: When discussing how to sort the word problems, students can use sentence stems: I know this problem is solving for lateral surface area because. I know this problem is solving for total surface area because. Extension for Learning: Use the information given to design a scale model of a pyramid. Find the surface area, and lateral surface area of the scale model and the actual pyramid. Information on the pyramids can be found at: Materials: chart paper scissors markers pencils anchors made in Making Connections activity Grade 8 STAAR Reference Chart Implementation Implementation: Put students in groups of 2 3. Pass out Surface Area Card Sort and chart paper Prompt students to make two columns on their chart paper (total surface area and lateral surface area) Have students complete the activity "Surface Area Sort." They should complete this in a Middle School Math

105 Surface Area Sort Supporting Activity group and sort them into the following categories: lateral surface area and total surface area. Have students check with you if their problems are correct. They should be able to tell you which clue words helped them determine whether the problem situation was asking for lateral or total surface area. Instruct students to glue their problems on the chart paper and solve them with their partners/group. Remember, for cylinders students should solve in terms of pi and then get the final answer as well. When groups are complete review problems as a class and select which posters will serve as anchors for the classroom. Begin extension activity, if time permits. Continue working with surface area with Carnegie Lesson 17.5 and skill practice Middle School Math

106 Surface Area Sort Supporting Activity Directions: Cut out the following problems. Next, decide if the problem is asking you to solve for lateral surface area or total surface area. Once you have grouped the problems, into the two categories, check your work with your teacher. Glue the problems in the correct categories on your poster paper and solve each problem. Maria is wrapping a birthday gift for a friend. The base of the box is a rectangle and has a length and width of 4 inches and 7 inches respectively. The height of the box is 6 inches. Crystal's parents are painting the walls of her bedroom for her birthday. Crystal s room measures 10.5 feet long, 7 feet wide, and 9 feet tall. How much paint will her parents need to buy to paint the walls of her bedroom? Mike wrapped a napkin around a soda can. If the soda can has a radius of 1.5 inches and a height of 4 inches, what is the amount of the can that the napkin covered? How much wrapping paper would she need to exactly cover the entire box with no extra paper? Mrs. Jackson made pizza dough with a rolling pin. The rolling pin has a radius of 1 inch and a height of 9 inches. Mr. Johnson is painting a room. He is using a paint roller. The radius of the roller is 2 inches and the length of the roller is 10 inches. How many square inches will his roller cover in one complete rotation? Audrey is using a soup can for a project. How much pizza dough will the rolling pin cover in two complete rotations? How many cubic inches of construction paper will she need to cover the entire surface of the soup can? Middle School Math

107 Surface Area Sort Supporting Activity Teacher Key Maria is wrapping a birthday gift for a friend. The base of the box is a rectangle and has a length and width of 4 inches and 7 inches respectively. The height of the box is 6 inches. Crystal's parents are painting the walls of her bedroom for her birthday. Crystal s room measures 10.5 feet long, 7 feet wide and 9 feet tall. How much paint will her parents need to buy to paint the walls of her bedroom? Mike wrapped a napkin around a soda can. If the soda can has a radius of 1.5 inches and a height of 4 inches, what is the amount of the can that the napkin covered? 12 or in 2 How much wrapping paper would she need to exactly cover the entire box with no extra paper? 188 in feet 2 Mrs. Jackson made pizza dough with a rolling pin. The rolling pin has a radius of 1 inch and a height of 9 inches. Mr. Johnson is painting a room. He is using a paint roller. The radius of the roller is 2 inches and the length of the roller is 10 inches. How many square inches will his roller cover in one complete rotation? Audrey is using a soup can for a project. 40 or in 2 How much pizza dough will the rolling pin cover in two complete rotations? 36 or in 2 How many cubic inches of construction paper will she need to cover the entire surface of the soup can? 52.5 or in 2 Middle School Math

108 Making Connections: Surface Area Foldable Activity Standards Content: 8.7B use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determining solutions for problems involving rectangular prisms, triangular prisms, and cylinders. ELPS: 3(E) share information in cooperative learning interactions 3(D) speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency Objectives Content Objective: Students will calculate surface area of rectangular prisms, triangular prisms and cylinders using nets and connect actions to the formulas on the STAAR 8 Reference chart. Language Objective: Students will be able to explain the relationship the 2-D figures that make up a 3- D figure and connect the relationship to the surface area formulas. Vocabulary Vocabulary: lateral surface area, total surface area, nets, cylinder, rectangular prism, triangular prism, perimeter, base, face Vocabulary Activity: Mix-N-Quiz 1. Each student is given a card that has a word from the Word Wall with its definition and a few facts. 2. Students stand with their card and find a partner. 3. Student A reads the definition and student B chooses a word from the Word Wall that matches the definition. 4. If student B does not choose the correct word, student A reads a fact and student B tries again. Middle School Math

109 Making Connections: Surface Area Foldable Activity Guiding Questions Is there only one way to calculate surface area? Why do you think part of the surface area formula involves finding the perimeter of the base and multiplying it by the height of the prism? Why do we add the area of the base times two? Differentiation English Language Learners: Use of anchors of support, sentence stems When using the formula, P means the of the base. The of a figure is a flat pattern that can be folded to form a 3-D shape. To calculate surface area, you can use the OR add up all the faces of a. Implementation Materials: scissors pencils/pens highlighters glue sticks triangular prism net, rectangular prism net, cylinder net Grade 8 STAAR Reference Chart Implementation: Students will cut the shapes out and only glue down one of the bottom faces so that the shape can be folded up to look like a solid figure, as pictured below. Note: The cylinder net should be glued down on one of the bases as well. Middle School Math

110 Making Connections: Surface Area Foldable Activity Class should work together to count the centimeter squares and add up all the shapes to get total surface area. See use in interactive notebook below. After each student has counted the centimeter squares and found the total surface area for each figure. Distribute a Grade 8 STAAR Reference chart to each student and discuss the formula for total surface area. Have students write the corresponding formula next to each net. Go through these questions together as a class. You can be making an anchor chart as well. o What does the P stand for? Lead students to highlight the perimeter on their net and the P in the formula. o Where is the height of the prisms? Highlight in a different color on the net and on the h in the formula. o What does the B stand for? Lead students to shade in the area of the base on their net and the B in the formula. o What is the perimeter of a circle called? Where is this on your net? Circumference. Lead students to highlight the circumference on the net and in the 2 r in formula. o Where is the height of the cylinder? Highlight in a different color on the net and on the h in the formula. o Where is the area of the base of your cylinder? Shade the area of the base and the B Middle School Math

111 Making Connections: Surface Area Foldable Activity in formula. Discuss the difference between lateral and total surface area. Lateral surface area is when the bases are not included. Lead student to understand that the lateral surface area on their nets are the areas in yellow and pink (or whatever two colors they used) and that this is the first part of each formula. Allow students to use these anchors in their notebooks and the classroom anchor as they work surface area problems in this CRM. Continue this topic with Carnegie Lesson 17.2 and skill practice Middle School Math