Introduction to the Third Dimension

Transcription

1 Introduction to the Third Dimension Rice is central to the daily diet of billions of people around the world. Ornamental containers, such as the ceramic canister shown, are common in kitchens where rice is cooked Cut, Fold, and Voila! Nets More Cans in a Cube The Cube Prisms Can Improve Your Vision! Prisms Outside and Inside a Prism Surface Area and Volume of a Prism The Egyptians Were on to Something or Was It the Mayans? Pyramids And The Winning Prototype Is...? Identifying Geometric Solids. in Everyday Occurrences

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3 Cut, Fold, and Voila! Nets Learning Goals In this lesson, you will: Sketch various views of a solid figure to provide a two-dimensional representation of a three-dimensional figure. Construct a net from a model of a geometric solid. Construct a model of a geometric solid from a net. Use nets to provide two-dimensional representations of a geometric solid. Key Terms geometric solids prototype edge face vertex net Have you ever heard of the term rebranding? Generally, this term means to give a product or an item a new look. Rebranding isn t a decision that businesses take lightly. Many times, marketing research is performed on a product s current look and possible new looks. There is also the risk that people will not recognize the product, perhaps leading to fewer sales. What items or products have you seen that have gone through rebranding? Do you think rebranding only deals with products or items? 14.1 Nets 905

4 Problem 1 Prototype #1 Geometric solids are all bounded three-dimensional geometric figures. The three dimensions are length, width, and height. The new marketing director of a rice distribution company, Rice Is Nice, has decided to change the way its product is packaged. The marketing director hopes to get more people to notice and talk about the product. She assigned her product development team to create prototypes. A prototype is a working model of a possible new product. Each prototype needs to be a different-shaped container to package the product. Rice is Nice is considering changing the dimensions of its current packaging. The box shown is one prototype. Prototype #1 The height of the box is 5.7 centimeters, the width or depth of the box is 2.9 centimeters, and the length of the box is 4.3 centimeters. 1. Use the figure shown to answer each question. a. How many sides of the box can you see? b. Describe the location of the sides you can see. c. How many sides can you not see? d. What sides can you not see? e. What is the shape of each side? 906 Chapter 14 Introduction to the Third Dimension

5 2. Sketch each side of the box, label the location of the side, and include the measurements. Imagine cutting out each side you sketched and taping the corresponding edges together to construct the box of Prototype #1. An edge is the intersection of two faces of a three-dimensional figure. A face is one of the polygons that makes up a polyhedron. The point where edges meet is known as a vertex of a three-dimensional figure. A vertex of a solid is similar to the vertex of an angle. A net is a two-dimensional representation of a three-dimensional geometric figure. A net is cut out, folded, and taped to create a model of a geometric solid. A net has all these properties: The net is cut out as a single piece. All of the sides of the geometric solid are represented in the net. The sides of the geometric solid are drawn such that they share common edges. The common edges are labeled as fold lines. Tabs are drawn on the edges to be taped Nets 907

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7 3. Cut, fold, and tape this net to create Prototype #1. Prototype # Nets 909

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9 Problem 2 Prototype #2 Sandy created this net to model her prototype for rice packaging. Cut out, fold, and tape this net to create a prototype. Prototype # Nets 911

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11 Problem 3 Prototype #3 Emilia created this net to model her prototype for rice packaging. Cut out, fold, and tape this net to create a prototype. Prototype # Nets 913

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13 Problem 4 Prototype #4 Trang created this net to model his prototype for rice packaging. Cut out, fold, and tape this net to create a prototype. Prototype # Nets 915

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15 Problem 5 Using Your Sorting Hat! The marketing director requires the team members to present their prototypes at the next Rice Is Nice stockholders meeting. She told the team that 4 prototypes are too many. The team members could not decide which of the prototypes to exclude, so they intend to group the 4 prototypes into 2 categories and highlight each category. 1. Using all 4 solids, sort them into 2 groups, and explain your reasoning. 2. Compare your method of grouping with your classmates. Did everyone use the same groupings? Explain your reasoning. Hang on to your prototype models, you will use these again in other lessons in this chapter Nets 917

16 The net for each prototype you created is shown. You will use these representations again in this chapter. 4.3 cm Prototype #1 2.9 cm 5.7 cm Prototype #2 5.7 cm 4.3 cm 4.3 cm 918 Chapter 14 Introduction to the Third Dimension

17 Prototype #3 5.3 cm 3.8 cm 3.3 cm Prototype #4 5.1 cm 3.7 cm Be prepared to share your solutions and methods Nets 919

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19 More Cans in a Cube The Cube Learning Goals In this lesson, you will: Create a model of a cube from a net. Construct a model of a geometric solid from a net. Use nets to provide two-dimensional representations of a cube. Estimate the volume and surface area of a cube. Use nets to compute the volume and surface area of a cube. Use a formula to determine the volume of a cube. Use unit cubes to estimate the surface area and volume of larger cubes. Use appropriate units of measure when computing the surface area and volume of a cube. Key Terms point line segment polygon polyhedron regular polyhedron congruent cube unit cube diameter surface area volume Explore how doubling the dimensions of a cube affects the volume of the cube. Only 90 years ago, the standard beverage size bottle was 6.5 ounces. Now, some convenient stores tout 128-ounce drinks! But the amount in one serving isn t the only thing that has gotten bigger. Why do you think drink portions have become larger over the years? Do you think the common practice of restaurants refilling drinks is a contributing factor? 14.2 The Cube 921

20 Problem 1 Speaking a Common Language Before beginning the lesson, everyone must speak a common language. It is important to use the same words when studying mathematics and describing geometric terms. For example, a word you may have used in the past may actually have a more precise definition when dealing with mathematics. For example, the word point has many meanings outside of math. However, the mathematical definition of point is a location in space. A point has no size or shape, but it is often represented by using a dot and is named by a capital letter. A line segment is a portion of a line that includes two points and all the points between those two points. Recall, a polygon is a closed figure formed by three or more line segments. Knowing these definitions will help you learn the meanings of other geometric words. 1. What do you think is the meaning of a closed figure? 2. Sketch what you think is an example of a polygon. 3. Is your sketch a closed figure? Are all of the sides in your sketch formed by line segments? 4. Compare your sketch with your classmates sketches. Did everyone sketch the same polygon? Explain how your classmates and your sketches are the same or different. 922 Chapter 14 Introduction to the Third Dimension

21 A polyhedron is a three-dimensional figure that has polygons as faces. 5. Sketch what you think is an example of a polyhedron. Would any of the prototypes you created in the last lesson be polyhedrons? 6. Does your sketch look like a three-dimensional figure? Does your sketch show polygons for every face? 7. Compare your sketch with your classmates sketches. Did everyone sketch the same polyhedron? Explain how the sketches are the same or different The Cube 923

22 A regular polyhedron is a three-dimensional solid that has congruent regular polygons as faces and has congruent angles between all faces. Congruent means having the same size, shape, and measure. 8. Sketch what you think is an example of a regular polyhedron. 9. Does your sketch look like a three-dimensional solid that has congruent regular polygons as faces and congruent angles between all the faces? 10. Compare your sketch with your classmates sketches. Did everyone sketch the same regular polyhedron? Explain how the sketches are the same or different. A cube is a regular polyhedron whose six faces are congruent squares. A unit cube is a cube that is one unit in length, one unit in width, and one unit in height. In this chapter, unit cubes are used as manipulatives to explore characteristics of geometric solids. The unit cubes are typically 1 centimeter in length, 1 centimeter in width, and 1 centimeter in height. For this reason, use a centimeter ruler to measure lengths in this chapter. 924 Chapter 14 Introduction to the Third Dimension

23 Problem 2 Is It Really a Cube? In 1993, beverage manufacturers decided to repackage their product to boost sales. Research indicated that consumers would rather buy more cans of their favorite beverages at one time than make several trips to the store. The marketing team came up with the idea of packaging several cans of their beverage together in a way that was easy to carry. This packaging is called the cube. A cube contains 24 cans. 1. Sketch some of the possible rectangular arrangements of 24 cans. Your arrangements may have more than one layer. The diameter of each can is 2 inches. The diameter is the distance across a circle through its center. The height of each can is 6 inches. 2. What are the approximate dimensions of rectangular boxes needed to contain each arrangement of cans you sketched in Question 1? 14.2 The Cube 925

24 3. The manufacturer decided to go with a two layer arrangement and called it a cube. a. What are the dimensions of this arrangement? b. Why do you think they made the decision to call this a cube? c. Explain why calling the package a cube can be confusing. 926 Chapter 14 Introduction to the Third Dimension

25 Problem 3 Characteristics of a Cube The cube is a basic geometric solid. 1. Sketch a cube. 2. How many faces of the cube can you see? 3. Describe the location of the faces you can see. 4. How many faces can you not see? 5. Describe the location of the faces that you cannot see. 6. What is known about the length, height, and width of the cube? 14.2 The Cube 927

26 7. Would you measure the length, width, and height of a cube using linear units such as inches, centimeters, and feet? Or, would you use square units such as square inches, square centimeters, and square feet? Or, would you use cubic units such as cubic inches, cubic centimeters, and cubic feet? 8. Sketch and describe the shape of each face of a cube. 9. Is a cube a polygon? Explain your reasoning. 10. Is a cube a polyhedron? Explain your reasoning. 928 Chapter 14 Introduction to the Third Dimension

27 Problem 4 Cube Net There are 11 different nets that can be created to model a cube. 1. Here is one example of a net of a cube. Cut it out, fold it, and tape it together to create a geometric model of a cube The Cube 929

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29 2. Describe the number of faces, vertices, and edges of the cube. 3. Not all of these nets create a cube. Circle each figure that is a net of a cube. Remember, a cube has six faces_and only six faces! 4. How did you determine which nets were cubes in Question 3? 14.2 The Cube 931

30 Problem 5 Surface Area of a Cube Surface area is the total area of the two-dimensional surfaces (faces and bases) that make up a three-dimensional object. Consider the model of the cube you created in Problem 4, Question 1, to answer each question. 1. What is true about the area of each of the 6 faces of a cube? 2. Is the area of a face of a cube measured using linear units such as inches, centimeters, and feet? Or, using square units such as square inches, square centimeters, and square feet? Or, using cubic units such as cubic inches, cubic centimeters, and cubic feet? 3. Describe a strategy that you can use to determine the total surface area of a cube? 4. Use a centimeter ruler to calculate the total surface area of your cube. 5. How is the net of a cube helpful when determining the surface area of a cube? 932 Chapter 14 Introduction to the Third Dimension

31 6. What is the surface area of a unit cube? 7. How are unit cubes helpful when determining the surface area of a larger cube? Do you have unit cubes ready? Problem 6 Volume of a Cube Volume is the amount of space occupied by an object. The volume of a cube is calculated by multiplying the length times the width times the height of the cube. Use the model of the cube you created in Problem 4, Question 1, to answer each question. 1. Would you measure the volume of a cube using linear units such as inches, centimeters, and feet? Or, would you use using square units such as square inches, square centimeters, and square feet? Or, would you use using cubic units such as cubic inches, cubic centimeters, and cubic feet? 2. How is estimating the volume of a cube different from calculating the volume of a cube? 14.2 The Cube 933

32 3. Estimate the volume of your cube by stacking unit cubes next to the model of the cube. Grab a handful of unit cubes! 4. Measure the length, width, and height of the cube using a centimeter ruler. Then, multiply the length, width, and height to calculate the volume. 5. What is the difference between the estimation of the volume and the calculation of the volume? What percent difference is considered a good estimate? 6. What is the ratio of the difference between the estimation and the calculation of the volume to the calculation of the volume? 7. Write the ratio from Question 6 as a percent. This is the percent of increase or decrease in volume resulting from estimation. 8. How is the net of a cube helpful when determining the volume of a cube? 9. What is the volume of a unit cube? 10. How could unit cubes be helpful when you are determining the volume of a larger cube? 934 Chapter 14 Introduction to the Third Dimension

33 Problem 7 Volume Formula Volume can be determined by using the formula V 5 3 w 3 h, where V is the volume of the cube, is the length of the cube, w is the width of the cube, and h is the height of the cube. The base of a cube is a square. Recall that the area of a square is calculated by multiplying the length of the square by the width of the square. Written as a formula, the area of the base of a cube is Area of the Base 5 3 w. Consider the two formulas: V 5 3 w 3 h Area of the Base 5 3 w If B is used to represent the area of the base of a cube, then you can rewrite the second formula as: B 5 3 w. Now consider the two formulas: V 5 3 w 3 h B 5 3 w Using both of these formulas, you can rewrite the formula for the volume of a cube as V 5 B 3 h, where V represents the volume of the cube, B represents the area of the base of the cube, and h represents the height of the cube. You can use this formula to calculate the volume of many different geometric solids. However, the formula for calculating the value of B will change depending on the base shape of the polyhedron The Cube 935

34 Use the formula V 5 B 3 h to answer each question. 1. The length, width, and height of the cube are each equal to 2 centimeters. 2 cm 2 cm 2 cm a. Calculate the area of the base of the cube. b. What is the height of the cube? Keep in mind that a number doesn't say much without a label. c. Calculate the volume of the cube. 2. The volume of this cube is 27 cubic centimeters. 27 cm 3 a. What is the area of the base of the cube? b. What is the height of the cube? 936 Chapter 14 Introduction to the Third Dimension

35 Problem 8 Jerome and Roberta Need Your Help! Jerome began stacking unit cubes to make a larger cube but was interrupted before he could finish. The figure shown displays how much progress Jerome made in making the larger cube. 1. You can see how long, wide, and tall Jerome wanted the cube. Calculate the volume and surface area of Jerome s cube if he had completed it. Roberta is using unit cubes to determine the surface area and volume of larger cubes. She wants to build 6 different size cubes to compare the surface area and volume, but she realized that she would not have enough unit cubes to complete the models. She decides to just build the length, width, and height of the first four cubes and to look for a pattern. The figures Roberta built are shown The Cube 937

36 2. Roberta thinks she sees a pattern, but she needs to sketch the fifth and sixth cube to be sure. Help Roberta by sketching the fifth and six figures based on the pattern you see. 3. Roberta is organizing the data in a table. Help Roberta complete the table. Dimensions of the Cube Area of One Side of the Cube (in square units) Surface Area of the Cube (in square units) Volume of the Cube (in cubic units) Describe how Roberta can use the dimension (length or width) of a cube to determine the area of one side of the cube. 5. Describe how Roberta can use the area of one side of the cube to determine the surface area of the cube. 938 Chapter 14 Introduction to the Third Dimension

37 6. Describe how Roberta can use the dimensions (length, width, and height) of the cube to determine the volume of the cube. 7. Roberta is looking at the completed table and notices that when the dimensions of a cube are doubled, the volume of the larger cube is predictable. She saw the pattern! Describe the pattern Roberta sees in the completed table. 8. Use Roberta s pattern and the volume of a unit cube to predict the volume of a unit cube. 9. Use Roberta s pattern and the volume of a unit cube to predict the volume of a unit cube. 10. Roberta s lab partner, Derrick, looked at her completed table in Question 3 and found it interesting that the surface area and the volume of a unit cube is 216. Roberta helped Derrick understand that they were not equal. What did Roberta tell Derrick? 14.2 The Cube 939

38 Talk the Talk Each numerical answer describes the volume or the surface area of a cube. Which is it? How do you know? Labels really do matter m m 3 Be prepared to share your solutions and methods. 940 Chapter 14 Introduction to the Third Dimension

39 Prisms Can Improve Your Vision! Prisms Learning Goals In this lesson, you will: Sketch a model of a right rectangular prism. Create models of various prisms. Determine the characteristics of various prisms. Key Terms prism bases of a prism lateral faces height of a prism rectangular prism right prism Do you know that binoculars and prisms are close friends? In 1854, Ignazio Porro realized this and patented the Porro Prism. Using right triangular prisms he was able to turn an image right-side up. Basically, when an image is gathered by the lens, the image is upside down. Thus, inside each eyepiece are two prisms, which turn the image right-side up and invert the image from left to right. What do you think would happen if only one prism was used in binoculars? What do you think would happen if 4 prisms were used in binoculars? 14.3 Prisms 941

40 Problem 1 Getting to Know Prisms! A prism is a polyhedron with two parallel and congruent faces and all the other faces are parallelograms. These two parallel and congruent faces are known as the bases of a prism. The remaining parallelogram-shaped faces are known as lateral faces. 1. What do you think two parallel and congruent faces means? 2. What is a parallelogram? 3. Sketch what you think is an example of a prism. Look at the prototype models you created earlier for ideas. 4. Does your sketch show two bases that are the same polygon and are they drawn parallel to each other? Are all of the faces of your polyhedron parallelograms except for the bases? 5. Identify the bases in your sketch. 942 Chapter 14 Introduction to the Third Dimension

41 6. Compare your sketch with your classmates sketches. Did everyone sketch the same prism? Explain how the sketches are the same or different. A height of a prism is the length of a line segment that is drawn from one base to the other base. This line segment must be perpendicular to the other base. 7. Use your sketch to explain what is meant by height of a prism? A rectangular prism is a prism that has a rectangle as its base. 8. Sketch what you think is an example of a rectangular prism. 9. Are all of the faces of your sketch rectangles? Does your sketch have two bases that are parallel and congruent? 10. Compare your sketch with your classmates sketches. Did everyone sketch the same rectangular prism? Explain how the sketches are the same or different Prisms 943

42 A right prism is a prism that has bases aligned one directly above the other and has lateral faces that are rectangles. All prisms associated with this chapter are right prisms. 11. Which faces of a prism are considered lateral faces? 12. Sketch what you think is an example of a right prism. 13. Are the bases of your prism aligned one directly above the other? Are all of the lateral faces in your sketch rectangles? 14. Compare your sketch with your classmates sketches. Did everyone sketch the same right prism? Explain how the sketches are the same or different. 944 Chapter 14 Introduction to the Third Dimension

43 Problem 2 Right Rectangular Prism 1. Use the model of the cube you created in Lesson 14.2 to answer each question. a. Is a cube a prism? Explain your reasoning. b. Is a cube a rectangular prism? Explain your reasoning. Can a square be a rectangle? Can a rectangle be a square? Ah! My head is spinning! c. Is a cube a right prism? Explain your reasoning. 2. In your own words, describe what makes up a right rectangular prism Prisms 945

44 3. Sketch a right rectangular prism that is not a cube. Have you created any other model that is a right rectangular prism? a. How many faces can you see? b. Describe the location of the faces you can see. c. How many faces can you not see? d. Describe the location of the faces you cannot see. e. What is known about the length, width, and height of the right rectangular prism? f. Describe the shape of each face of the right rectangular prism. 946 Chapter 14 Introduction to the Third Dimension

45 Problem 3 Characteristics of a Prism Use pasta and miniature marshmallows to construct a model of each prism shown. Use your model to answer questions about the prisms. 1. Construct and analyze this prism. You can break the pasta to be any length you want. a. Name the polygon that is the base of this prism. b. How many faces of the prism are lateral faces? c. Identify the number of vertices, edges, and faces. d. How is a height of this prism determined? 14.3 Prisms 947

46 2. Construct and analyze this prism. a. Name the polygon that is the base of this prism. b. How many faces of the prism are lateral faces? c. Identify the number of vertices, edges, and faces. d. How is a height of this prism determined? 948 Chapter 14 Introduction to the Third Dimension

47 3. Construct and analyze this prism a. Name the polygon that is the base of this prism. b. How many faces of the prism are lateral faces? c. Identify the number of vertices, edges, and faces. d. How is a height of this prism determined? 14.3 Prisms 949

48 4. Construct and analyze this prism. a. Name the polygon that is the base of this prism. b. How many faces of the prism are lateral faces? c. Identify the number of vertices, edges, and faces. d. How is a height of this prism determined? 950 Chapter 14 Introduction to the Third Dimension

49 5. Complete the table shown with the data from Questions 1, 2, 3, and 4. Shape of the Base of Prism (Regular Polygon) Number of Sides of the Base Number of Vertices Number of Edges Number of Faces 6. Use the data from the table in Question 5 and any patterns you notice to answer each question. a. What is the relationship between the number of sides of the base and the number of vertices of each prism? b. What is the relationship between the number of sides of the base and the number of edges of each prism? c. What is the relationship between the number of sides of the base and the number of faces of each prism? 7. Without making a model or drawing a sketch, predict the number of vertices, edges, and faces for an octagonal prism. Describe your reasoning for making the prediction. 8. To verify your prediction, make a model of an octagonal prism to check your answers. Be prepared to share your solutions and methods Prisms 951

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51 Outside and Inside a Prism Surface Area and Volume of a Prism Learning Goals In this lesson, you will: Use unit cubes to estimate the volume and surface area of a right rectangular prism. Use nets to compute the volume and surface area of a right rectangular prism. Use a formula to determine the volume of a right rectangular prism. Use appropriate units of measure when computing the surface area and volume of a right rectangular prism. Many zoos and aquariums that hold large marine animals must order or build custom-made tanks for their creatures. These tanks come in many different shapes and sizes. A large fish tank that might hold jellyfish or sea horses at the zoo might have dimensions of 48" 24" 25" and could hold around 115 gallons of water. Tanks for large animals such as dolphins and whales could have dimensions of 46' 23' 30'. That means these tanks hold around 238,000 gallons of water! How do you think the zoo keepers determine what size the tank should be and how much water it should hold? 14.4 Surface Area and Volume of a Prism 953

52 Problem 1 Right Rectangular Prism Net 1. The net shown is Prototype 1 from the first lesson. 2.9 cm 5.7 cm 4.3 cm Prototype #1 Get out your prototype 1 model to help answer questions in this lesson. a. Write the name of each side on each face: front, back, top, bottom, left side, and right side. Pairs of faces in the prism are congruent. This can help me estimate. b. Use the net to estimate the surface area of the right rectangular prism. Recall that the unit of measurement when calculating the surface area is square units. c. Calculate the surface area of the right rectangular prism. Explain your calculation. 954 Chapter 14 Introduction to the Third Dimension

53 d. How does the estimation of the surface area compare to the calculation of the surface area? e. Use your model of a right rectangular prism to determine the number of faces, vertices, and edges. f. Estimate the maximum number of unit cubes that would fit inside your model of a right rectangular prism. Calculating the actual volume of a right rectangular prism is similar to calculating the actual volume of a cube. Multiply the length of the rectangular prism times the width of the rectangular prism times the height of the rectangular prism, or calculate the area of the base and multiply the product by the height. g. Calculate the volume of the right rectangular prism. Recall that the unit of measurement when calculating the volume is cubic units. h. How does the estimation of the volume compare to the calculation of the volume? 14.4 Surface Area and Volume of a Prism 955

54 2. Place your model of the right rectangular prism you created on your desk such that it rests on one of the largest sides. a. Lightly sketch a letter X on the two bases of the prism. b. Now, turn your model of the right rectangular prism such that it rests on one of the smallest sides and lightly sketch a letter X on the two bases of the prism. c. Are there two Xs on any face of the prism? d. Think of the front, back, left, right, bottom, and top faces as locations on the prism. Is the location of the bases in part (a) the same location as the bases in part (b)? e. In your own words, describe how you can determine which sides of a right rectangular prism are bases. 956 Chapter 14 Introduction to the Third Dimension

55 Problem 2 Surface Area of a Prism The surface area of the cube was calculated by adding all of the areas of all of the faces of the cube. The same process is also used to determine the surface areas of prisms. Similar to the cube, the area of a base of a rectangular prism can be calculated using the area formula, B 5 3 w, where B is the area of the base, is the length of the rectangular base, and w is the width of the rectangular base. However, many of the prisms in this lesson do not have rectangular bases. The base of a prism can be a variety of polygons. If the base of the prism is a pentagon, a hexagon, an octagon, or any polygon different from a rectangle, you need to use a strategy to determine the area of the base. One base of a regular pentagonal prism is shown. Recall that when a polygon is regular, that means all of the sides of the polygon are equal in length and all of the angles of the polygon are equal in measure. 1. Locate and place a point at the center of the pentagon. From the center point, draw line segments to connect the point with each vertex of the pentagon. 2. Describe the new polygons formed by adding these line segments. 3. What information do you need to calculate the area of each new polygon? 4. What formula is used to calculate the area of each new polygon? 5. Describe a strategy to determine the area of the entire pentagonal base Surface Area and Volume of a Prism 957

56 One base of a regular hexagonal prism is shown. 6. Use the same strategy you used in Question 1 to divide the hexagon into new polygons. 7. Describe the new polygons formed by adding these line segments. 8. What information do you need to calculate the area of each new polygon? 9. What formula is used to calculate the area of each new polygon? 10. Describe a strategy to determine the area of the entire hexagonal base. 11. Do you think this strategy works for any regular polygonal base of a prism? Explain your reasoning. In conclusion, the surface area of a prism is the sum of the areas of all of the lateral faces of the prism plus the areas of the two bases. 958 Chapter 14 Introduction to the Third Dimension

57 Problem 3 Volume Formula of a Prism The formula for calculating the volume of a cube is V 5 B 3 h, where V represents the volume of the cube, B is the area of the base of the cube, and h is the height of the cube. The same is true for prisms. Use the same formula, but apply it to a prism. 1. What does the variable V represent? 2. What does the variable B represent? 3. What does the variable h represent? 4. Write the formula for determining the volume of a prism. Define all variables used in the formula. 5. Describe the strategy used for determining the area of the base of the prism when the base is a regular polygon but not rectangular Surface Area and Volume of a Prism 959

58 Talk the Talk Each numerical answer describes the volume or the surface area of a right rectangular prism. Which is it? How do you know? cm 2 cm 1.4 cm cm 5.2 cm 1.25 cm m m Chapter 14 Introduction to the Third Dimension

59 Prisms are named by the shape of their bases. 5. Name the polygons that best describe the bases of each prism. a. a pentagonal prism b. an octagonal prism c. a triangular prism d. a decagonal prism e. a hexagonal prism f. a heptagonal prism 14.4 Surface Area and Volume of a Prism 961

60 6. Use the nets shown to determine the name of each prism. a. b. c. d. e. f. Be prepared to share your solutions and methods. 962 Chapter 14 Introduction to the Third Dimension

61 The Egyptians Were on to Something or Was It the Mayans? Pyramids Learning Goals In this lesson, you will: Create a model of a pyramid from a net. Use nets to provide two-dimensional representations of a pyramid. Use nets to estimate the surface area of a pyramid. Use appropriate units of measure when computing the surface area of a pyramid. Key Terms pyramid vertex of a pyramid height of a pyramid slant height of a pyramid The Great Pyramids of Egypt are a favorite tourist spot for any travelers in the area. Not to be outdone, the pyramids of Mexico are also a favorite tourist attraction and quite challenging to climb! Egypt does have what is considered to be the oldest pyramid in the world. The Step Pyramid in Saqqara is considered to be the oldest stone pyramid. Some experts date the pyramid was built between 2649 and 2575 BC! How do you think archaeologists determine the age of a structure? Do you think there was a reason why two civilizations were alike in building pyramids? 14.5 Pyramids 963

62 Problem 1 Getting to Know Pyramids A pyramid is a polyhedron with one base and the same number of triangular faces as there are sides of the base. The triangular faces are called lateral faces. The vertex of a pyramid is the point at which all lateral faces intersect. All of the pyramids associated with this chapter have a vertex that is located directly above the center point of the base of the pyramid. Don't pyramids have other vertices also? 1. Sketch the first thing that comes to your mind when you hear the word pyramid. 2. Does your sketch have one base? Does your sketch have the same number of triangular faces as there are sides of the base? 3. Identify the vertex of the pyramid on your sketch. 4. Compare your sketch with your classmates sketches. Did everyone sketch the same pyramid? Explain how the sketches are the same or different. 964 Chapter 14 Introduction to the Third Dimension

63 Similar to a triangle, a height of a pyramid is the length of a line segment drawn from the vertex of the pyramid to the base. This line segment is perpendicular to the base. 5. Use your sketch to explain what is meant by the height of a pyramid. A slant height of a pyramid is the distance measured along a lateral face from the base to the vertex of the pyramid along the center of the face. As shown, a slant height, s, is the altitude of a triangular lateral face of the pyramid. Do you notice the right angle symbol where the slant height touches the base? S The height of a triangular face of a pyramid is a dimension often needed to calculate the total surface area of the pyramid Pyramids 965

64 Problem 2 Characteristics of a Pyramid Use pasta and miniature marshmallows to construct a model of each pyramid. Use your model to answer questions about the pyramids. 1. Construct and analyze this pyramid. Just like last time, you can break the pasta to any length you want. a. Name the polygon that is the base of this pyramid. b. How many faces of the pyramid are lateral faces? c. Describe the intersection of all of the lateral faces. d. How many vertices, edges, and faces are in your model? e. How can you determine the height of your pyramid? 966 Chapter 14 Introduction to the Third Dimension

65 2. Construct and analyze this pyramid. a. Name the polygon that is the base of this pyramid. b. How many faces of the pyramid are lateral faces? c. Describe the intersection of all of the lateral faces. d. How many vertices, edges, and faces are there? e. How can you determine the height of your pyramid? 14.5 Pyramids 967

66 3. Construct and analyze this pyramid. a. Name the polygon that is the base of this pyramid. b. How many faces of the pyramid are lateral faces? c. Describe the intersection of all of the lateral faces. d. How many vertices, edges, and faces are there? e. How can you determine the height of this pyramid? 968 Chapter 14 Introduction to the Third Dimension

67 4. Construct and analyze this pyramid. a. Name the polygon that is the base of this pyramid. b. How many faces of the pyramid are lateral faces? c. Describe the intersection of all of the lateral faces. d. How many vertices, edges, and faces are there? e. How can you determine the height of your pyramid? 14.5 Pyramids 969

68 5. Organize the data from Questions 1, 2, 3, and 4 by completing the table shown. Shape of the Base of Pyramid (Regular Polygon) Number of Sides of the Base Number of Vertices Number of Edges Number of Faces 6. Use the table you completed in Question 5 to answer each question. a. What is the relationship between the number of sides of the base and the number of vertices of each pyramid? b. What is the relationship between the number of sides of the base and the number of edges of each pyramid? c. What is the relationship between the number of sides of the base and the number of faces of each pyramid? 7. Without making a model or drawing a sketch, predict the number of vertices, edges, and faces for an octagonal pyramid. Describe your reasoning for making your prediction. 8. To verify your prediction, make a model of an octagonal pyramid to check your answers. 970 Chapter 14 Introduction to the Third Dimension

69 Problem 3 Pyramid Net Mr. Morris instructed his math students to use their straw-and-marshmallow models of a square pyramid to help them create a net. Shawna raised her hand and said that she had an idea. She said that all she had to do was remove the marshmallow that was at the top of the pyramid, lower the straws that formed the lateral sides, and reuse the marshmallow somewhere else, but she would need 3 additional marshmallows to complete the net. 1. Sketch Shawna s net of a square pyramid. Explain why she would need 3 additional marshmallows Pyramids 971

70 2. Mr. Morris instructed his students to create a model of a pentagonal pyramid. Then, he wanted them to create a net from their model. If Shawna uses the same strategy she used to create the net for the square pyramid, how many additional marshmallows will she need to build a net for a regular pentagonal pyramid? 3. Create Shawna s net for a regular pentagonal pyramid. 972 Chapter 14 Introduction to the Third Dimension

71 4. Allen raised his hand and claimed that he created a different regular pentagonal pyramid net. A drawing of Allen s net is shown. How many additional marshmallows will Allen need? 14.5 Pyramids 973

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73 Problem 4 Surface Area of a Pyramid Use Allen s net of a regular pentagonal pyramid to estimate the surface area of a pyramid. 1. What information would you need to estimate the area of one triangle in Allen s net? 2. Describe a strategy to estimate the area of the base of the pyramid in Allen s net. 3. Use Allen s net and a centimeter ruler to estimate the surface area of the pyramid. Recall that the unit of measurement when estimating surface area is square units Pyramids 975

74 4. Shawna is not convinced that Allen s net is a pyramid because it looks different than hers. Help to convince Shawna by cutting out, folding, and taping Allen s net to show it forms a regular pentagonal pyramid. 5. Do you think the strategy used to calculate the surface area of the regular pentagonal pyramid also work for pyramids that have different regular polygonal bases? You have just used various strategies to calculate surface area. However at this point, you will not generate a formula for determining the surface area of a pyramid. You will explore the surface area of a pyramid in depth when you study geometry in high school. 976 Chapter 14 Introduction to the Third Dimension

75 Talk the Talk 1. Solve for the surface area of each pyramid. a. A square pyramid where the length of each side of the base is 10 inches and the slant height is also 10 inches. b. The base of the pyramid is a regular pentagon. 14 cm 3.5 cm 6 cm Like prisms, pyramids are named by the shape of their bases. 2. Name the polygon that best describes the base of each pyramid. a. a pentagonal pyramid b. an octagonal pyramid c. a triangular pyramid d. a decagonal pyramid e. a hexagonal pyramid f. a heptagonal pyramid 14.5 Pyramids 977

76 3. Use the nets to determine the name of each pyramid. a. b. c. d. e. f. Be prepared to share your solutions and methods. 978 Chapter 14 Introduction to the Third Dimension

77 And The Winning Prototype Is...? Identifying Geometric Solids in Everyday Occurrences Learning Goals In this lesson, you will: Identify geometric solids. Compare and contrast the surface area of geometric solids. Apply the surface area concept to a real-world situation. When was the last time you saw a circle? Or perhaps, when was the last time you saw a line in the geometric terms? In fact, when you begin to formally study geometry in high school, most of your instruction will begin with two-dimensional figures; however, the world is full of three-dimensional objects. Even a piece of paper may appear to be a two-dimensional object, but it isn t! It does have a depth, even though that depth is quite small. Why do you think that most geometry courses start with two-dimensional examples? Do you think there are some principles that are key in two-dimensional examples that will be used when studying three-dimensional objects? 14.6 Identifying Geometric Solids in Everyday Occurrences 979

78 Problem 1 Geometric Solids are Everywhere! 1. Geometric solids appear in real life in a variety of places. Identify each solid. a. b. c. d. e. f. 980 Chapter 14 Introduction to the Third Dimension

79 g. h. i. j. k. l. m. n Identifying Geometric Solids in Everyday Occurrences 981

80 Problem 2 Gathering Information Throughout this chapter, you have estimated and calculated the volume and surface area of four prototypes developed by the Rice Is Nice product development team. It is now time to compile this information and develop a business plan to market each prototype. The amount of money it costs the manufacturers to package a product and the amount of money generated by the sale of this product will determine the profit margin. 1. Complete the table with the information of each prototype for Rice Is Nice. Prototype Number Name of the Geometric Solid Surface Area (in cm 2 ) Prototype #1 Prototype #2 Prototype #3 Prototype #4 The nets representing each of the prototypes can be found at the end of the first lesson in this chapter. 982 Chapter 14 Introduction to the Third Dimension

81 2. Match each sketch with the appropriate prototype number Identifying Geometric Solids in Everyday Occurrences 983

82 3. Consumers are concerned about how much of the product they get for their money. If the cost of each rice container is the same, does the measurement of the surface area help them to determine the best buy? Explain. How are surface area and volume related? 4. Manufacturers are concerned with maximizing their profit. Does the measurement of the surface area help them to determine the best choice? Explain. Problem 3 Commemorative Canisters The product development team members came up with a great idea to introduce their new rice container. They decided to give consumers a complimentary commemorative metal canister with their first purchase of the newly packaged product. The metal canister will maintain the same size and same shape as the prototype container. The stockholders of the Rice Is Nice Company asked the development team to calculate the cost of materials used to make the commemorative canisters for the four prototypes. The team wants to compare the price of using aluminum, tin, and copper. 1. Calculate the cost of using aluminum. One rectangular sheet of aluminum measuring 25.4 centimeters long, 10.2 centimeters wide, and 0.04 centimeters thick will cost $2.69. a. How many square centimeters are in one rectangular sheet of aluminum? b. Determine the cost of aluminum per square centimeter to the nearest tenth of a cent. When calculating an amount of money, always round up. 984 Chapter 14 Introduction to the Third Dimension

83 c. Use the information from Problem 2 to complete the table. Prototype Number Name of the Geometric Solid Surface Area (cm 2 ) Cost of Aluminum (dollars) Prototype #1 Prototype #2 Prototype #3 Prototype #4 2. Calculate the cost of using tin. One rectangular sheet of tin measuring 25.4 centimeters long, 10.2 centimeters wide, and 0.02 centimeters thick will cost $3.09. a. How many square centimeters are in one rectangular sheet of tin? b. Determine the cost of tin per square centimeter to the nearest tenth of a cent. When calculating an amount of money, always round up. c. Use the information from Problem 2 to complete the table. Prototype Number Name of the Geometric Solid Surface Area (cm 2 ) Cost of Tin (dollars) Prototype #1 Prototype #2 Prototype #3 Prototype # Identifying Geometric Solids in Everyday Occurrences 985

84 3. Calculate the cost of using copper. One rectangular sheet of copper measuring 25.4 cm long, 10.2 cm wide, and 0.06 cm thick will cost $9.49 a. How many square centimeters are in one rectangular sheet of copper? b. Determine the cost of copper per square centimeter to the nearest tenth of a cent. When calculating an amount of money, always round up. c. Use the information from Problem 2 to complete the table. Prototype Number Name of the Geometric Solid Surface Area (cm 2 ) Cost of Copper (dollars) Prototype #1 Prototype #2 Prototype #3 Prototype #4 986 Chapter 14 Introduction to the Third Dimension

85 Talk the Talk Team up with a few classmates to write a report to the director of marketing. In the report, recommend one of the three prototypes for production, and the material that should be used to produce the commemorative canister. Explain your reasoning. Then, present your report to the class and the class can decide which report is the most convincing. Be prepared to share your solutions and methods Identifying Geometric Solids in Everyday Occurrences 987

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87 Chapter 14 Summary Key Terms geometric solids (14.1) prototype (14.1) edge (14.1) face (14.1) vertex (14.1) net (14.1) point (14.2) line segment (14.2) polygon (14.2) polyhedron (14.2) regular polyhedron (14.2) congruent (14.2) cube (14.2) unit cube (14.2) diameter (14.2) surface area (14.2) volume (14.2) prism (14.3) bases of a prism (14.3) lateral faces (14.3) height of a prism (14.3) rectangular prism (14.3) right prism (14.3) pyramid (14.5) vertex of a pyramid (14.5) height of a pyramid (14.5) slant height of a pyramid (14.5) Constructing a Net from a Model of a Geometric Solid Geometric solids are bounded three-dimensional geometric figures. The three dimensions are length, width, and height. A net is a two-dimensional representation of a geometric solid. A net can be cut out, folded, and glued or taped to create a model of a geometric solid. When constructing a net, it may be helpful to make a sketch of each side or face of the geometric solid first. Then, connect each side in such a way that they share common edges. When folded along these edges, the net should be a model of the geometric solid. Example Whoo! That was a tough chapter and I know I made a lot of mistakes but you know, as Einstein said, "A person who never made a mistake, never tried anything new." A net is sketched from the given model of a geometric solid. Chapter 14 Summary 989

88 Notice how the faces that share sides in the net share common edges in the model of the cube. The other faces that share edges in the model are connected by the tabs. top left front right back bottom Calculating the Surface Area and Volume of Cubes A polyhedron is a 3-dimensional solid that has polygons as faces. A regular polyhedron has congruent regular polygons as faces and has congruent angles between all faces. A cube is a regular polyhedron whose six faces are congruent squares. Surface area is the total area of the 2-dimensional surfaces that make up a 3-dimensional object. The surface area of a cube is calculated by determining the area of one face and then multiplying that area by 6. Volume is the amount of space occupied by an object. To calculate the volume of a cube, use the formula V 5 B 3 h, where V represents the volume of the cube, B represents the area of the base of the cube, and h represents the height of the cube. Example To determine the area of one face of the cube, multiply the length times the width. Area of the base: cm 2 7 cm Surface area of the cube: cm 2 Use the formula V 5 B 3 h to calculate the volume of the cube. V 5 B 3 h The volume of the cube is 343 cm 3. When a cube s dimensions are doubled, the volume of the resulting cube is eight times the volume of the initial cube. If the given cube s dimensions were doubled, the volume would be cm Chapter 14 Introduction to the Third Dimension

89 Calculating the Surface Area and Volume of Right Rectangular Prisms A prism is a polyhedron with two parallel and congruent faces called bases. All other faces are parallelograms and are referred to as lateral faces. A rectangular prism is a prism that has rectangles as its bases. A right prism is a prism that has bases aligned one directly above the other and has lateral faces that are rectangles. To calculate the surface area of a right rectangular prism, calculate the area of each rectangular face and add each of these areas together. As with cubes, the volume of a prism can be determined by using the formula V 5 B 3 h, where V represents the volume of the prism, B represents the area of the base of the prism, and h represents the height of the prism. Example 2 cm 3 cm 7 cm Surface area of the right rectangular prism: 2(7 3 2) 1 2(7 3 3) 1 2(2 3 3) cm 2 Area of the base of the prism: cm 2 Use the formula V 5 B 3 h to calculate the volume of the prism. V 5 B 3 h The volume of the prism is 42 cm 3. Chapter 14 Summary 991