Surface Area and Volume of Prisms

Transcription

1 CHAPTER 12 Surface Area and Volume of Prisms connected.mcgraw-hill.com Investigate Animations Vocabulary Math Songs The BIG Idea How are threedimensional solids alike and different? How do I find the surface area and volume of prisms? Multilingual eglossary Learn Personal Tutor Virtual Manipulatives Make this Foldable to help you organize information about perimeter, area, and volume. Audio Three - Dimensional Figures Surfacece Area Volume Foldables Practice Self-Check Practice egames Worksheets Assessment Review Vocabulary Polygon polígono a closed figure made up of line segments that do not cross each other. Key Vocabulary English Español prism prisma surface area área total three-dimensional figure figura tridimensional volume volumen 590

2 When Will I Use This? Your Turn! You will solve this problem in the chapter. Surface Area and Volume of Prisms 591

3 Are You Ready for the Chapter? You have two options for checking Prerequisite Skills for this chapter. Text Option Take the Quick Check below. Tell whether each pair of figures is congruent or similar Describe the lines as parallel, perpendicular, or neither Tell whether each shape is a polygon Online Option Take the Online Readiness Quiz. 592 Surface Area and Volume of Prisms

4 Multi-Part Lesson 1 Properties of Three-Dimensional Figures PART A B C D E Main Idea I will build nets and explore properties of three-dimensional figures. Build Three-Dimensional Figures A net is a two-dimensional pattern of a three-dimensional figure. You can use a net to build a three-dimensional figure. A three-dimensional figure has length, width, and height. Vocabulary net Materials colored pencils A vertex is a point where 3 or more edges meet. A face is a flat side. An edge is where two faces meet. grid paper Step 1 Copy the net shown onto grid paper. scissors Get ConnectED GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Identify a three-dimensional object from two-dimensional representations of that object and vice versa. Also addresses GLE Step 2 Congruent shapes have the same size and shape. Shade all of the congruent shapes the same color. Step 3 Cut out the net. Fold it along the solid black lines to form the three-dimensional figure. You have formed a cube. About It 1. What two-dimensional figure forms the faces of a cube? How many faces are there? How many are congruent? 2. How many edges and vertices (the plural of vertex) are there? Lesson 1A Properties of Three-Dimensional Figures 593

5 Step 1 Copy the net shown onto grid paper. Step 2 Shade the congruent shapes the same color. Step 3 Cut out the net. Fold it along the solid black lines to form the three-dimensional figure. You have formed a rectangular prism. Step 1 Copy the net shown onto grid paper. Step 2 Shade the congruent shapes the same color. Step 3 Cut out the net. Fold it along the solid black lines to form the three-dimensional figure. You have formed a triangular prism. 594 Surface Area and Volume of Prisms

6 Step 1 Copy the net shown onto grid paper. Step 2 Shade the congruent shapes the same color. Step 3 Cut out the net and fold it along the solid black lines to form the three-dimensional figure. You have formed a rectangular pyramid. About It 3. How many faces, vertices, and edges are there in the figure in Activity 2? 4. In Activity 3, what two-dimensional figures form the faces of the triangular prism? How many faces are congruent? 5. How many edges and vertices are there in the figure in Activity 4? and Apply It Create a three-dimensional figure from each net. Then describe the faces, edges, and vertices Which three-dimensional figure below has the most vertices? most edges? A B C D 9. E WRITE MATH Is there more than one way to create a net for a cube? Explain. Lesson 1A Properties of Three-Dimensional Figures 595

7 Multi-Part Lesson PART 1 Properties of Three-Dimensional Figures A B C D E Main Idea I will analyze properties of three-dimensional figures. Vocabulary three-dimensional figure polyhedron face edge vertex prism base pyramid cone cylinder Get ConnectED Three-Dimensional Figures Recall that a two-dimensional figure is a plane figure that has length and width. A three-dimensional figure has length, width, and height. A three-dimensional figure with faces that are polygons is called a polyhedron. A face is a flat side. An edge is where two faces meet. A vertex is a point where 3 or more edges meet. Two types of three-dimensional figures are prisms and pyramids. They are named by the shape of their bases. GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Given a series of geometric statements, draw a conclusion about the figure described. Prisms A prism has at least three faces that are rectangles. The top and bottom faces, called the bases, are congruent parallel polygons. Rectangular Triangular Square prism prism prism or cube A rectangular prism has six rectangular faces and eight vertices. A triangular prism has five faces and six vertices. A cube has six square faces and eight vertices. 596 Surface Area and Volume of Prisms

8 Characteristics of Solids Parallel same distance apart and never intersects Congruent same size and shape Describe the faces, edges, and vertices of the three-dimensional figure. Then identify it. faces edges vertices This figure has 5 faces. The triangular bases are congruent and parallel. The other faces are rectangles. There are 9 edges. The edges that form the vertical sides of the rectangles are parallel and congruent. This figure has 6 vertices. The figure is a triangular prism. Pyramids A pyramid has at least three faces that are triangles. It has only one base, which is a polygon. Triangular pyramid Rectangular pyramid CAMPING Describe the faces, edges, and vertices of the tent. Then identify the shape of the tent. faces The tent has 5 faces. There is one base that is a rectangle. The other 4 faces appear to be triangles. edges vertices There are 8 edges. The opposite edges of the base are parallel and congruent. The tent has 5 vertices. The tent is a rectangular pyramid. Lesson 1B Properties of Three-Dimensional Figures 597

9 Some three-dimensional figures have curved surfaces. Cones and Cylinders Cones A cone has only one base. The base is a circle. Has one vertex. Cylinders A cylinder has two bases. The bases are congruent circles. Has no vertices and no edges. SPORTS Describe the faces, edges, and vertices of the container. Then identify the shape of the container. faces Characteristics of Solids The circular bases are congruent. They are perpendicular to the curved surface of the container. edges vertices The container has no edges. The container has no vertices. So, the container is a cylinder. Describe the faces, edges, and vertices of each three-dimensional figure. Then identify it. See Examples E 3. TALK MATH Describe the differences between a cylinder and a rectangular prism. 598 Surface Area and Volume of Prisms

10 PRACTICE EXTRA Begins on page EP2. Describe the faces, edges, and vertices of each three-dimensional i figure. Then identify it. See Examples What kind of three-dimensional figure is the tomato soup can at the right? 13. Describe the number of vertices and edges in an unopened cereal box. Identify the shape of the box. 14. Vera s closet is in the shape of a rectangular prism. Describe the pairs of parallel sides that make up her closet. 15. WHICH ONE DOESN T BELONG? Which figure does not belong with the other three? Explain your reasoning. 16. REASONING Which three-dimensional figure has 4 faces, 4 vertices, and 6 edges? 17. CHALLENGE What figure is formed if only the height of a cube is increased? Draw the figure to support your answer. 18. E WRITE MATH Describe the similarities and differences of a rectangular prism and a triangular prism. To assess mastery of SPI , see your Tennessee Assessment Book. 599

11 Multi-Part Lesson 1 Properties of Three-Dimensional Figures PART A B C D E Main Idea I will draw orthogonal and projective views of three-dimensional figures. Vocabulary orthogonal view projective view Get ConnectED Views of Three- Dimensional Figures You can draw different views of three-dimensional figures. The orthogonal view of a three-dimensional figure is the top, side, and front views of a figure. A projective view of a threedimensional figure shows the picture view of the figure. Projective View SPI Identify a three-dimensional object from two-dimensional representations of that object and vice versa. Top View Side View Front View Orthogonal View Draw the Orthogonal View of a Figure The projective view of a threedimensional figure is shown. Draw the top, side, and front views of the figure. The top view shows 2 rows of 4. The side view resembles a reverse L. The front view also shows 2 rows of Surface Area and Volume of Prisms

12 Draw the Projective View of a Figure The orthogonal view of a three-dimensional figure is shown. Draw the projective view of the figure. Top View Side View Front View Use the top view to build the figure s base. top Use the side and front views to complete the figure. side front Draw the top, side, and front view of each figure PRACTICE EXTRA Begins on page EP2. Draw the projective view of each figure Top View Side View Front View Top View Side View Front View Draw the top, side, and front view of each object To assess mastery of SPI , see your Tennessee Assessment Book. 601

13 Multi-Part Lesson PART 2 Surface Area of Prisms A B C D E Surface Area of Prisms Main Idea I will explore using models to find the surface area of rectangular prisms. Suppose you want to paint all of the surfaces of the prism. You would need to find the surface area of this prism. To find the surface area, you add the areas of all the faces of the prism. Materials grid paper Create a net to find the surface area of the prism. Step 1 Draw and cut out the net below. scissors Step 2 Fold along the solid black lines. Tape the edges together to form a prism. Step 3 Find the area of each of the six faces of the prism. Get ConnectED top GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Solve problems involving surface area and volume of rectangular prisms and polyhedral solids. Also addresses GLE side Face front bottom side back front and back top and bottom two sides Model Area (units 2 ) Step 4 Find the sum of the areas. A = A = 126 units 2 Surface area has square units because it measures area. 602 Surface Area and Volume of Prisms

14 Find the surface area of the rectangular prism. 5 ft top 7 ft 7 ft 7 ft 5 ft 4 ft back side front side 4 ft 4 ft 7 ft 5 ft bottom 7 ft 5 ft Find the area of each face. Then add. Face front and back top and bottom two sides Model Area (ft 2 ) A = or 166 square feet About It 1. Describe the faces of a cube. Explain how to find the surface area of a cube. and Apply It Make a net to find the surface area of each rectangular prism m 4 m 3 m 5 ft 12 ft 2 ft 4 in. 4 in. 4 in. 5. E WRITE MATH How many pairs of congruent faces are in a rectangular prism? Describe them. Lesson 2A Surface Area of Prisms 603

15 Multi-Part Lesson 2 PART Surface Area of Prisms A B C D E Main Idea I will find the surface area of rectangular prisms. Vocabulary surface area Surface Area of Prisms The sum of the areas of all the faces of a prism is called the surface area of the prism. Each face of a rectangular prism has a congruent opposite face. So, the following formula can also be used to find surface area. Get ConnectED Surface Area of a Rectangular Prism GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Solve problems involving surface area and volume of rectangular prisms and polyhedral solids. Also addresses GLE Words Model To find the surface area of a rectangular prism, add the areas of all the faces of the prism. h w l h w side l back bottom front side h top l Symbols S.A. = 2lh + 2lw + 2hw Find the Surface Area GIFTS Find the surface area for the amount of wrapping paper needed to cover the gift. Find the area of each face. top and bottom: 2lw = or 36 9 in. front and back: 2lh = or 108 two sides: 2wh = or 54 Add to find the surface area. 6 in. 3 in. The surface area is or 198 square inches. 604 Surface Area and Volume of Prisms

16 Area is the number of square units needed to cover a region. It is measured in square units. CAMERAS Digital cameras are made small enough to fit in a pocket. This camera is shaped like a rectangular prism. Find the surface area of the camera. Find the area of each face. top and bottom: 2lw = or 24 front and back: 2lh = or 48 two sides: 2wh = or 16 Add to find the surface area. 6 in. The surface area is or 88 square inches. 2 in. 4 in. Find the surface area of each rectangular prism. See Examples 1 and ft 11 mm 3 ft 9 ft 7 mm 12 mm 15 in. 6 in. 2 in. 4. A box of animal crackers is shaped like a rectangular prism. What is the surface area of the box of crackers? 4 cm 5 cm 5. Find the surface area of a rectangular prism with a length of 9 meters, a width of 7 meters, and a height of 4 meters. 7 cm E 6. TALK MATH The formula for the surface area of a rectangular prism is S.A. = 2lw + 2lh + 2wh. Explain why there are three 2s in the formula. Lesson 2B Surface Area of Prisms 605

17 Find the surface area of each rectangular prism. See Examples 1 and in. 6 cm 3 in. 14 in. 4 cm 8 cm PRACTICE EXTRA Begins on page EP2. 12 mm 12 mm 12 mm ft m 7 ft 2 ft 9 in. 18 m 6 m 4 in. 5 in. 13. Alyssa owns a toolbox that is 16 inches by 22 inches by 5 inches. What is the surface area of the toolbox? 14. Michelle put her sister s birthday present in a box with a length of 13 mm, a width of 4 mm, and a height of 8 mm. How many square millimeters of wrapping paper will Michelle need to completely cover the box? 2 in. 15. A package of three golf balls comes in the box shown. What is the surface area of the box? 7 in. 16. Which has a greater surface area: a box that is 2 inches by 3 inches by 2 inches or a box that is 1 inch by 2 inches by 4 inches? 2 in. 17. CHALLENGE What is the possible length, width, and height of a rectangular prism with the surface area of 110 square centimeters? 18. OPEN ENDED Estimate the surface area of a cereal box. Then measure and find the actual surface area. Compare to the estimate. 19. E WRITE MATH Explain how to find the surface area of a cube without using the formula. 606 Surface Area and Volume of Prisms

18 Test Practice 20. What is the surface area of the box of hot chocolate? A. 210 i n 2 B. 216 i n 2 C. 325 i n 2 6 in. 22. SHORT RESPONSE If the area of the top of the figure shown is 16 square centimeters, what is the area of the bottom? D. 340 i n 2 10 in. 3 in. 21. For a science project, Madison uses foil to cover the outside of a can. She does not cover the top or the bottom of the can. What two-dimensional figure represents the shape of the piece of foil that Madison uses? F. circle G. hexagon H. rectangle I. triangle 23. Which statement is true about the figure? A. The figure has a triangular base. B. The figure has exactly 3 pairs of parallel faces. C. The figure has exactly 2 pairs of parallel faces. D. The figure has 7 vertices. Describe the faces, edges, and vertices of each three-dimensional figure. Then identify it. (Lesson 1B) Cara made the candle shown for her mother. Describe the faces, edges, and vertices of the candle. Then identify it. To assess partial mastery of SPI , see your Tennessee Assessment Book. 607

19 Get Ready! Players: 2 Get Set! Decide who will be Player 1 and Player 2. GO! 10 Questions Three-Dimensional Figures Player 1 will start by writing the name of a three-dimensional figure on a piece of paper. If Player 2 guesses the correct figure within the 10 questions, then he/she earns 1 point. Alternate asking and guessing between the two players. The first person with 5 points wins. You Will Need: paper and pencil Player 2 will ask questions about the figure. For example: Does this figure have 5 faces? Does this figure have less than 6 vertices? Are there 9 edges in this figure? Player 1 will respond to Player 2 s questions with yes or no. Player 2 will ask up to 10 questions about the three-dimensional figure. 608 Surface Area and Volume of Prisms

20 Mid-Chapter Check Create a three-dimensional figure from each net. Then describe the faces, edges, and vertices. (Lesson 1A) MULTIPLE CHOICE Nelly gave her mother the present shown. Find the surface area of the present. (Lesson 2B) 5 in. 14 in. 2. Describe the faces, edges, and vertices of each three-dimensional figure. Then identify it. (Lesson 1B) in. F. 140 square inches G. 260 square inches H. 520 square inches I. 600 square inches Find the surface area of each rectangular prism. (Lesson 2B) 9. 3 m in m 1 m 2 in. 6 in. 7. MULTIPLE CHOICE Which h statement is true about the figure? (Lesson 1B) ft 4 ft 4 ft cm 15 cm 5 cm A. The figure has 4 vertices. B. The figure has 2 circular bases. C. The figure has a triangular base. D. The figure has 1 circular base. 13. The dimensions of a jewelry box are 7 inches by 4 inches by 5 inches. Find the surface area of the jewelry box. 14. E WRITE MATH Draw and label a cube with a surface area of 150 square units. Mid-Chapter Check 609

21 Multi-Part Lesson 3 Volume of Prisms PART A B C D E Problem-Solving Strategy: Make a Model Main Idea I will solve problems by making a model. Nick is helping his younger sister put away her alphabet blocks. She has already put away one layer of blocks. To fill up one layer, it takes nine blocks. If the box is filled with six layers of blocks, how many blocks would be in the box? A B C Understand What facts do you know? The number of blocks in one layer of a box. The number of layers in the box. What do you need to find? The number of blocks in the box when there are six layers. Plan Solve the problem by making a model. Solve Use your plan to solve the problem. Make a model of one layer of the box by arranging 9 cubes in a 3 3 array. Continue stacking the cubes until there are six layers. There are a total of 54 cubes. So, the box would have 54 blocks. Check Look back. Use logical reasoning and multiplication. There are 6 layers and each layer has 9 cubes. So, the total number of cubes is 6 9 or 54. The answer is correct. GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Solve problems involving surface area and volume of rectangular prisms and polyhedral solids. Also addresses GLE Surface Area and Volume of Prisms

22 Refer to the problem on the previous page. 1. How many blocks would the box contain if it had only five layers of blocks? 2. List the length, width, and height of two different boxes that can hold exactly 54 blocks with none left over. 3. What are the advantages of the make a model strategy? 4. List some objects that you could use to make a model. PRACTICE EXTRA Begins on page EP2. Solve. Use the make a model strategy. 5. Measurement On an assembly line that is 150 feet long, there is a work station every 15 feet. The first station is at the beginning of the line. How many work stations are there? 6. A store is stacking cans of food into a pyramid-shaped display. The bottom layer has 9 cans. There are 5 layers. If there are two less cans in each layer, how many cans are in the display? 7. Measurement The distance around the center ring at the circus is 80 feet. A clown stands every 10 feet along the circle. How many clowns are there? 8. Measurement Martino wants to arrange 18 square tiles into a rectangular shape with the least perimeter possible. How many tiles will be in each row? 9. In the figure below, there are 22 marbles in Box A. To go from Box A to Box B, four marbles must pass through the triangular machine at a time. Five marbles must pass through the square machine at a time. Describe how to move all the marbles from Box A to Box B in the fewest moves possible. 10. Drake lined up 15 pennies on his desk. He replaced every third penny with a nickel. Then he replaced every fourth coin with a dime. Finally, he replaced every fifth coin with a quarter. What is the value of the remaining 15 coins on his desk? Explain. 11. E WRITE MATH Describe when you would use the make a model strategy. Lesson 3A Volume of Prisms 611

23 Multi-Part Lesson 3 PART Volume of Prisms A B C D E Volume of Prisms Main Idea I will use models to find the volumes of prisms. You can use centimeter cubes to build rectangular prisms like the ones shown at the right. Materials centimeter cubes Step 1 Use centimeter cubes to build four different rectangular prisms. Get ConnectED GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Solve problems involving surface area and volume of rectangular prisms and polyhedral solids. Step 2 For each prism, record the dimensions and the number of cubes used in a table like the one below. Prism A B C D Length (l) Width (w) Height (h) Number of Cubes Since volume can be measured using cubes, volume is measured in cubic units or unit s 3. and Apply It 1. Describe the relationship between the dimensions of the prism and number of cubes. 2. Use l, w, and h to write a formula for the volume V of a rectangular prism. 3. Use the formula you wrote in Exercise 2 to find the volume of the prism at the right in appropriate units. Verify your solution by counting the number of cubes. 612 Surface Area and Volume of Prisms

24 Multi-Part Lesson 3 Volume of Prisms PART A B C D E Main Idea I will find the volumes of rectangular prisms. Vocabulary volume Get ConnectED GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Solve problems involving surface area and volume of rectangular prisms and polyhedral solids. Also addresses GLE , GLE Volume of Prisms Volume is the amount of space inside a three-dimensional figure. Volume is measured in cubic units. A cubic unit has length, width, and height. 1 cubic unit 2 cubic units 4 cubic units You can find the volume of a rectangular prism by using models. A cube with 1 unit on an edge is a standard unit for measuring volume. When cubes are placed in a prism to determine volume, there are no gaps or overlaps between the cubes. MAIL Evelyn wants to mail a package to her cousin. What is the volume of the package if it is 6 inches long, 4 inches wide, and 4 inches tall? Count the number of 1-inch cubes that will fill the bottom 4 in. of the rectangular prism. The prism is 6 cubes long and 4 cubes wide. There are 4 in. 24 cubes on the bottom. 6 in. There are 4 layers of cubes. So, there are 4 24 or 96 cubes. 4 in. 6 in. 4 in. Lesson 3C Volume of Prisms 613

25 Some common units of volume are cubic inch, cubic foot, cubic yard, cubic centimeter, and cubic meter. Volume measurements can be written using abbreviations and an exponent of 3. For example: cubic units = unit s 3 cubic inches = i n 3 cubic feet = f t 3 cubic meters = m 3 The volume of a rectangular prism is related to its dimensions. You can use a formula to find the volume of a prism. Volume of a Rectangular Prism Words To find the volume of a Model rectangular prism multiply the length, width, and height. h Symbols V = lwh l w Volume of a Prism ART Armando makes sand paintings by filling clear plastic cases with colored sand. Find the volume of the plastic case. Estimate = 125 V = lwh Formula for volume V = l = 5, w = 4, h = 7 V = 140 Multiply. The volume of the prism is 140 cubic inches. 4 in. 5 in. 7 in. Check for Reasonableness Volume of a Prism Find the volume of the prism. Estimate = 1,000 V = lwh Formula for volume V = l = 12, w = 9, h = 10 V = 1,080 Multiply. The volume of the prism is 1,080 c m 3. Check for Reasonableness 1,080 1, cm 9 cm 10 cm 614 Surface Area and Volume of Prisms

26 Find the volume of each prism. See Examples m 4 in. 4 m 6 m 4 in. 4 in. 9 cm 2 cm 5 cm 4. l = 21 cm, w = 8 cm, h = 4 cm 5. l = 19 ft, w = 9 ft, h = 16 ft 6. Find the cubic feet of air in a room that is 13 feet long, 10 feet high, and 11 feet wide. E 7. TALK MATH Describe which units would be appropriate to measure the volume of a jewelry box. What other units might be reasonable to use? Would it be reasonable to use the same units to measure the volume of a garage? Explain. Find the volume of each prism. See Examples PRACTICE EXTRA Begins on page EP2. 3 cm 11 ft 12 cm 3 ft 30 ft 26 cm m 11 in. 23 m 11 in. 11 in. 9 m in. 9 cm 17 cm 7 cm 2 in. 4 in. Lesson 3C Volume of Prisms 615

27 Find the volume of each prism. See Examples l = 5 yd, w = 16 yd, h = 6 yd 15. l = 2 m, w = 8 m, h = 10 m 16. l = 13 in., w = 3 in., h = 2 in. 17. l = 13 cm, w = 8 cm, h = 10 cm 18. Find the volume of a bank vault that is 14 feet by 20 feet by 19 feet. Use the information to solve the problem. 19. Determine the volume of each pet carrier. Which one should Emma purchase? 20. OPEN ENDED Estimate the volume of a shoe box. Then measure the box. Check your estimate by finding the actual volume. 21. NUMBER SENSE Describe the dimensions of two different prisms that have a volume of 2,400 cubic centimeters. 22. CHALLENGE A store sells lunch boxes that measure 11 inches by 7 inches by 4 inches. How many lunch boxes will fit in a box that measures 22 inches by 15 inches by 8 inches? Explain. 4 in. 11 in. 7 in. 8 in. 22 in. 15 in. 23. E WRITE MATH Write a real-life problem that could be solved by finding the volume of a prism. Then solve. 616 Surface Area and Volume of Prisms

28 Test Practice 24. Popcorn tins are stacked in a display so that there are 12 tins in the bottom row. There are 10 tins in the next row, and 8 tins in the row above that. There are five rows of tins. If the pattern continues, how many popcorn tins are there in all? A. 22 C Shawn keeps his photos in a box like the one shown. B. 30 D SHORT RESPONSE Find the volume in cubic meters of a cube with the dimension shown. What is the volume in cubic inches of the box? A. 22 B. 72 C m D What is the volume of the rectangular prism shown below? 28. SHORT RESPONSE A rectangular prism made of 1-inch cubes is shown below. 5 cm 7 cm 3 cm F. 88 c m 3 H. 120 c m 3 What is the volume of the prism? G. 105 c m 3 I. 142 c m A box of crayons has a length of 9 inches, a width of 4 inches, and a height of 6 inches. What is the surface area of the box of crayons? (Lesson 2B) 30. What kind of three-dimensional shape is shown? (Lesson 1B) To assess partial mastery of SPI , see your Tennessee Assessment Book. 617

29 Multi-Part Lesson PART 3 Volume of Prisms A B C D E Main Idea I will select and use appropriate units and formulas to measure surface area and volume. Select Appropriate Measurement Formulas It is important to be able to choose the appropriate measurement for a given situation. Get ConnectED GLE Describe polyhedral solids and analyze their properties, including volume and surface area. SPI Solve problems involving surface area and volume of rectangular prisms and polyhedral solids. Measure Used to Find Formula Surface Area Volume Measurement Formulas For Rectangular Prisms area of all surfaces space enclosed by a figure S.A. = 2lh + 2lw + 2hw V = lwh TRUCKS A family wants to rent the truck that can hold the most. To determine which truck can hold the most, should you find the surface area or volume? Solve. Truck Sizes Length Width Height Truck A 10 ft 6 ft 6 ft Truck B 16 ft 7 ft 7 ft Truck C 24 ft 7 ft 7 ft You need to determine how much the truck can hold. So, you need to find the volume. Truck A or 360 cubic feet Truck B or 784 cubic feet Truck C or 1,176 cubic feet Truck C can hold the most. 618 Surface Area and Volume of Prisms

30 Surface area is given in square units, and volume is given in cubic units. DESIGN An interior designer needs to calculate how much wallpaper it will take to cover 4 decorative columns. The figure represents one column. Determine whether she should find the surface area or volume of the columns. Then solve. The interior designer needs to know how much surface to cover. So, she needs to find the surface area. S.A. = 2lh + 2lw + 2hw 6 m 1 m 1 m Formula for surface area of a rectangular prism S.A. = 2(1 6) + 2(1 1) + 2(6 1) l = 1, w = 1, and h = 6 S.A. = S.A. = 26 Simplify. Add. So, 26 square meters of wall paper are needed for one column. Multiply 26 by 4 to find the surface area of 4 columns = 104 The surface area of 4 columns is 104 m 2. Determine whether you need to find the surface area or volume. Then solve. See Examples 1 and 2 1. Shane is mailing a DVD player as a gift to his cousin. He wraps the gift in bubble wrap to help protect it. How much bubble wrap is needed? 2. The gift described in Exercise 1 will be mailed in a box that has a length of 30 inches, a width of 15 inches, and a height of 7 inches. How much space will the box enclose? 3. How much water is needed to fill a pool that is 50 meters long, 25 meters wide, and 3 meters deep? 4. Which units would be most appropriate to measure the volume of a cake pan: cubic inches, cubic feet, or cubic yards? Explain. 5. E TALK MATH Explain how you determine whether to use the formula for surface area or volume for a given situation. Lesson 3D Volume of Prisms 619

31 Determine whether you need to find the surface area or volume. Then solve. See Examples 1 and 2 6. A path is 3 feet wide and 14 feet long. How much gravel will Mr. James need if he wants to add 2 inches of gravel over the entire path? PRACTICE EXTRA Begins on page EP2. 7. A company packages sticks of butter to be sold in stores. How much wrapping does the company need for each stick of butter? 8. Beth is painting her dresser white. Her dresser is 3 feet high, 3 feet long, and 2 feet wide. Is Beth painting the surface area or volume? To help stay fit, some people have swimming pools. Swimming pools can come in many shapes and sizes. The table shows two examples of rectangular swimming pools. Swimming Pool Sizes Pool Length Width Height Pool A 15 m 10 m 2 m Pool B 20 m 15 m 2 m 9. If you wanted to fill a pool, would you find the surface area or volume? 10. How many cubic meters of water would fill Pool A? 11. How many cubic meters of water would fill Pool A and Pool B? 12. FIND THE ERROR Tom is finding the volume of a prism with length 3 meters, height 8 meters, and width 17 meters. Help find and correct his mistake. V = 3 8 = 24 cubic meters 13. E WRITE MATH Explain the difference between finding the surface area and the volume of a rectangular prism. 620 Surface Area and Volume of Prisms

32 Test Practice 14. Bonita is making a building out of boxes. She wants to cover a box with the dimensions shown below with brown paper. 15. Gabriel wants to determine how much the box shown can hold. 8 in. 14 in. 8 in. How much paper will it take to cover the entire box? A. 112 square inches B. 576 square inches C. 896 cubic inches D. 900 cubic inches Which formula should he use? F. A = lw G. S.A. = 2lh + 2lw + 2hw H. V = lwh I. A = _ 1 2 bh 16. Craig is painting his bedroom. The room has four walls that are each 9 feet long by 10 feet tall. A gallon of paint covers 120 square feet. How many gallons should he buy to cover all four walls? (Lesson 3A) Find the surface area of each rectangular prism. (Lesson 2B) in. 18 in. 9 in cm 6 cm 4 cm yd 9 yd 25 yd Describe the faces, edges, and vertices of each three-dimensional figure. Then identify it. (Lesson 1B) Lesson 3D Volume of Prisms 621

33 Multi-Part Lesson PART 3 Volume of Prisms A B C D E Problem-Solving Investigation Main Idea I will choose the best strategy to solve a problem. JACINDA: I have twelve cubes. There are many ways to arrange them to form a rectangular prism. I want to find the arrangement of cubes that will have the least surface area. YOUR MISSION: Find the arrangement of cubes that has the least surface area. understand You need to find the arrangement of cubes that has the least surface area. Plan Solve Solve the problem by making a table. Make a table listing possible ways 12 cubes can be arranged to form a rectangular prism with different surface areas. All prisms have a volume of 12 cubic units. Rectangular Prism Dimensions (units) Surface Area (units 2 ) (S.A. = 2lh + 2lw + 2hw) l = 3, h = 2, w = 2 32 l = 6, h = 2, w = 1 40 l = 4, h = 1, w = 3 38 l = 12, h = 1, w = 1 50 The rectangular prism with the least surface area would have a length of 3 units, a height of 2 units, and a width of 2 units. check Reread the problem to see if the answer matches the information given. 622 Surface Area and Volume of Prisms

34 Guess, check, and revise. Look for a pattern. Make a model. Make a table. Draw a diagram. PRACTICE EXTRA Begins on page EP2. 4. The volume of a rectangular prism is 5,376 cubic inches. The prism is 14 inches long and 16 inches wide. How tall is the prism? Use any strategy to solve each problem. 1. Geometry Mariana has 24 cubes. How can she stack the cubes to create a figure with the greatest possible surface area? 2. Measurement A hexagon that has each side equal to 1 inch has a perimeter of 6 inches. perimeter = 6 in. Two hexagons placed side by side have a perimeter of 10 inches. Three hexagons have a perimeter of 14 inches. perimeter = 10 in. perimeter = 14 in. What would be the perimeter of five hexagons placed side by side? 5. Algebra The table below shows the number of minutes Danielle spent practicing the trumpet over the last 7 days. If she continues this pattern of practicing, in how many days will she have practiced 340 minutes? Day Time (min) For Exercises 6 and 7, use the following information. Marita wants to make a rectangle with a perimeter of 20 inches. 6. How many rectangles can Marita make if she only uses whole numbers for the side lengths? List the dimensions. 7. Which rectangle has the greatest area? 3. Five friends are standing in a circle and playing a game where they toss a ball of yarn to one another. If each person is connected by the yarn to each other person only once, how many lines of yarn will connect the group? 8. E WRITE MATH One wall of a building is 80 feet long and 16 feet high. A one-gallon can of paint covers up to 450 square feet. If each can of paint costs $22.50, find the total cost of paint for the wall. Explain the steps you used to solve the problem. Lesson 3E Volume of Prisms 623

35 You probably eat packaged frozen food every day. Frozen food might seem like a simple concept, but there s more to it than just putting a container of food in the freezer. Clarence Birdseye is sometimes called the father of frozen food because he was the first to develop a practical way to preserve food by flash freezing. Birdseye experimented with freezing fruits and vegetables, as well as fish and meat. His method of freezing food preserved the food s taste, texture, and appearance. He also was the first to package food in waxed cardboard packages that could be sold directly to consumers. 148 patents were issued that related to Clarence Birdseye s flash-freezing method, his type of packaging, and the packaging materials he used. 624 Surface Area and Volume of Prisms

36 Dimensions of Frozen Food Packages in Inches Item Length Width Height Fish Sticks Hamburger Patties Pizza Vegetables Frozen Dinner Use the information above to solve each problem. 1. What is the volume of a frozen pizza 5. A larger package of frozen package? vegetables has the same length and width but twice the height. What is the volume of this package? 2. How much more space does a package of fish sticks occupy than a package of vegetables? 6. Use a centimeter ruler to measure the length, width, and height of an actual frozen food package to the nearest whole unit. Then find the surface area of the package. 3. Is 200 cubic inches a reasonable estimate for the volume of a frozen dinner package? Explain. 4. A freezer has 2,600 cubic inches of available space. After seven packages of hamburger patties are placed inside, how much available freezer space is left? 7. E WRITE MATH Explain the differences between surface area and volume and the units used to represent them. Problem Solving in Science 625

37 Chapter Study Guide and Review Three - Dimensional Figures Key Concepts Surface Area Volume Three-Dimensional Figures (Lesson 1) A three-dimensional figure has length, width, and height. A face is a flat side. An edge is where two faces meet. Surface Areas of Rectangular Prisms (Lesson 2) Be sure the following Big Ideas are written in your Foldable. A vertex is a point where 3 or more edges meet. The surface area S.A. of a rectangular prism is the sum of the areas of the faces. S.A. = 2lh + 2lw + 2hw Vocabulary cone cube cylinder prism pyramid rectangular prism square pyramid surface area three-dimensional figure triangular prism triangular pyramid volume Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. To find the surface area of a rectangular prism, multiply the length l by the width w by the height h. 2. A cube with all sides 4 centimeters long has a volume of 64 cubic centimeters. 3. A square pyramid has 6 vertices. Volumes of Prisms (Lesson 3) The volume V of a rectangular prism is length l times width w times height h. V = lwh 4. A three-dimensional figure with faces that are polygons is called a polyhedron. 5. A cylinder has two parallel congruent circular bases. 626 Surface Area and Volume of Prisms

38 Lesson 1 Properties of Three-Dimensional Figures Three-Dimensional Figures (Lesson 1B) Describe the faces, edges, and vertices of the three-dimensional figure. Then identify it EXAMPLE 1 Describe the faces, edges, and vertices of the three-dimensional figure. Then identify it. 8. Describe the faces, edges, and vertices of the vase. Then identify the shape of the vase. Opposite faces are parallel and congruent. There are 12 edges and 8 vertices. The figure is a rectangular prism. Lesson 2 Surface Area of Prisms Surface Area of Prisms (Lesson 2B) Find the surface area of each rectangular prism in. 8 in ft 10 ft 4 ft EXAMPLE 2 Find the surface area of the rectangular prism. S.A. = 2lh + 2lw + 2hw 8 cm 4 cm 5 cm 3 in. 11. A DVD player measures 17 inches by 15 inches by 3 inches. What is the minimum surface area of a box to hold the DVD player? front and back: 2lh = or 64 top and bottom: 2lw = or 80 two sides: 2hw = or 40 The surface area is or 184 square centimeters. Chapter Study Guide and Review 627

39 Chapter Study Guide and Review Lesson 3 Volume of Prisms Problem-Solving Strategy: Make a Model (Lesson 3A) Solve by making a model. 12. A box is filled with 48 cubes that measure 1 inch on each side. The cubes completely fill the box. What are possible dimensions of the box? 13. Haley is making a bracelet by placing beads and charms on a 6-inch chain. She places a charm at 1 inch from each end, and at every _ 1 inch in 2 between. How many charms does she use? 14. Destiny has 24 feet of fencing material to make a pet enclosure. Describe three different rectangular areas that can be enclosed by the fencing. EXAMPLE 3 How many centimeter cubes will fit in the container at the right? You can use cubes to model the situation. First, arrange 2 rows of 5 cubes. Next, add three more layers of cubes. The total number of cubes used is 40. So, 40 centimeter cubes will fit in the container. Volume of Prisms (Lesson 3C) Find the volume of each prism ft 2 ft 7 m 8 ft 5 m m in. 16 cm 9 cm 3 in. 22 cm 3 in. 19. Victoro keeps his pet rabbit in a cage that is shaped like a rectangular prism. The cage measures 2 feet by 3 feet by 2 feet. What is the volume of the cage? EXAMPLE 4 Find the volume of the prism. 10 m 15 m 8 m V = lwh Volume of a prism V = l = 15, w = 10, h = 8 V = 1,200 Multiply. The volume of the prism is 1,200 cubic meters. 628 Surface Area and Volume of Prisms

40 Lesson 3 Volume of Prisms (continued) Select Appropriate Measurement Formulas (Lesson 3D) Determine whether you need to find the surface area or volume. Then solve. 20. Julius wants to purchase the wastebasket that can hold the most. Which wastebasket should he purchase? Wastebasket Dimensions Basket Length (cm) Width (cm) Height (cm) A B C A safe is painted red on each side. How much paint do you need to cover the safe? 8 in. 4 in. EXAMPLE 5 How much space is inside the box shown? Determine whether you need to find the surface area or volume. Solve. You need to determine how much the box can hold. So, you need to find the volume. V = lwh Volume of a prism. V = l = 4, w = 2, and h = 3. V = 24 cubic feet EXAMPLE 6 How much plastic wrap will it take to cover the entire box? Determine whether you need to find the surface area or volume. Solve. 5 in. 22. A box has a length of 6 meters, a width of 4 meters, and a height of 7 meters. Could you fit 200 cubic meters of material inside of the box? Explain. 23. Kristin is wrapping a present. The present is in a box with a length of 9 centimeters, a width of 5 centimeters, and a height of 10 centimeters. How much wrapping paper will she need? 12 in. 9 in. 16 in. You need to find the surface area. S.A. = 2(lh) + 2(lw) + 2(hw) S.A. = 2(16 12) + 2(16 9) + 2(12 9) S.A. = 2(192) + 2(144) + 2(108) Multiply. S.A. = Multiply. S.A. = 888 square inches Add. Chapter Study Guide and Review 629

41 Chapter Study Guide and Review Lesson 3 Volume of Prisms (continued) Problem-Solving Investigation: Choose a Strategy (Lesson 3E) Use any strategy to solve. 24. Barrett has 18 sports cards. He collects football and baseball cards. He has twice as many baseball cards. How many of each kind does he have? 25. Leon has $5 to buy a bottle of water that costs $1.49, a granola bar for $1.09, and the magazine shown below. Does he have enough money? Explain. EXAMPLE 7 What two whole numbers have a sum of 12 and a product of 32? Understand You know that there are two numbers that added together equal 12. You also know the same two numbers, when multiplied by each other, equal 32. Plan Use the guess, check, and revise strategy. Solve Guess: 2 and Sarah went bowling with her family. She had 3 more strikes than her brother, but 1 less strike than her dad. If there were a total of 7 strikes, how many strikes did each person bowl? 27. Ian has $21.75 in his pocket after he purchased a DVD for $ Would $50, $60, or $70 be a reasonable estimate for the money in Ian s pocket before he bought the DVD? 28. The volume of a rectangular prism is 112 cubic inches. The prism has a length of 7 inches and a height of 8 inches. What is the width of the prism? Check: = 12, 2 10 = 20 Revise: Since 20 32, try 2 different numbers Guess: 3 and 9 Check: = 12, 3 9 = 27 Revise: Since 27 32, try 2 different numbers Guess: 4 and 8 Check: = 12, 4 8 = 32 The whole numbers are 4 and 8. Check Since = 12 and 4 8 = 32 the two numbers chosen are correct. 630 Surface Area and Volume of Prisms

42 Practice Chapter Test Describe the faces, edges, and vertices of each three-dimensional figure. Then identify it MULTIPLE CHOICE Which figure below has 3 more edges than faces? F. H. G. I If you place one cube on a table, you can see 5 faces of the cube. If you place a second cube on top of the first, you can see 9 faces. How many faces can you see in a stack of 6 cubes? 6. MULTIPLE CHOICE For an art project, Heather needs to cover the box shown with craft sticks. What is the surface area of the box? A. 17 in 2 C. 42 in 2 B. 28 in 2 D. 188 in 2 Find the surface area of each prism m 9 m 9 m 8. 5 ft 10 ft 3 ft Find the volume of each prism m 4 m 10 m in. 7 in. 7 in. For each problem, determine whether you need to find the surface area or volume. Then solve. 12. A design is printed on the outside of a tissue box. To determine how much ink you will need for the design, find the amount of space you need to cover. 6 cm 13. A storage shed is 13 feet long, 8 feet wide, and 10 feet tall. How much can the storage shed hold? 8 cm 5 cm 14. E WRITE MATH Describe the difference between finding the surface area of a rectangular prism and finding the volume of a rectangular prism. Practice Chapter Test 631

43 Test Practice Drew has a fish aquarium. How many cubic inches of water can the aquarium hold? 14 in. Before beginning a problem, you may need to determine if you should find the surface area or the volume. 20 in. 9 in. Read the Test Item You need to find the volume of the aquarium. Solve the Test Item V = lwh V = V = 2,520 in 3 The aquarium can hold 2,520 cubic inches of water Fill in the grid. Read each question. Then fill in the correct answer on the answer sheet provided by your teacher or on a sheet of paper. 1. Which of the following could be a volume measurement? A. 250 inches B. 250 square inches C. 250 in 2 D. 250 in 3 2. Which of the following would you use to find the surface area of the figure shown? F. S.A. = l w G. S.A. = 2l + 2w H. S.A. = lwh I. S.A. = 2lh 2lw 2hw 632 Surface Area and Volume of Prisms

44 3. How many faces, edges, and vertices does the figure have? 6. GRIDDED RESPONSE A rectangular driveway measures 16 feet by 12 feet. What is the area of the driveway in square feet? A. 5 faces, 8 edges, 5 vertices B. 5 faces, 6 edges, 8 vertices C. 5 faces, 8 edges, 6 vertices D. 6 faces, 10 edges, 6 vertices 4. Look at the pattern of numbers shown below. 7,, 17, 22, 27, 32 Which expression is equal to the missing number in the pattern? F. (27-18) + 3 G. (7 + 12) - 5 H. (17-12) + 5 I. (28-23) SHORT RESPONSE Derrick is painting a toy chest with the dimensions shown. Find the surface area and volume of the chest. 7. Paul plays a video game twice. He receives a score of points the first time and points the second time. How many more points did Paul score the second time? A. 50 C B D Which statement about the figures shown below is true? Q S F. Figures Q and R have opposite sides that are parallel and congruent. G. Figures S and T are squares. H. Figures S and R have the same number of vertices. I. Figures Q and T each have 5 edges. R T 14 in. 11 in. 9 in. 9. SHORT RESPONSE It takes 3 feet of wood to make 1 birdhouse. If you have 23 feet of wood, can you make 8 birdhouses? Explain. NEED EXTRA HELP? If You Missed Question Go to Chapter-Lesson C 12 2B 12 1B 9 1A 12 2B 11 2C 5 3C 12 1B 3 3B For help with... SPI 4.4 SPI 4.4 GLE 4.2 SPI 3.2 SPI 4.4 GLE 4.1 SPI 2.5 SPI 1.1 SPI 2.3 Test Practice 633