CCM6+ Unit 12 Surface Area and Volume page 1 CCM6+ UNIT 12 Surface Area and Volume Name Teacher Kim Li MAIN CONCEPTS Page(s) Unit 12 Vocabulary 2 3D Figures 3-8 Volume of Prisms 9-19 Surface Area 20-26 Unit 12 Review and Practice Test 27-31 Essential Standards 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in the contest of solving real-world and mathematical problems. 6.G4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems. Essential Questions What are the similarities and differences between faces, edges and vertices? What is the difference between volume and surface area? What are some misconceptions about finding the volume with fractional edge lengths? How would you determine the number of fractional cubes that would fit inside a shape? How is area related to volume? Surface area? When would you need to find volume or surface area in the real world? Give examples.
CCM6+ Unit 12 Surface Area and Volume page 2 polyhedron three-dimensional figure whose surfaces, or faces, are all polygons area the amount of square units covered by a plane figure measured in square units net an arrangement of two-dimensional figures that can be folded to form a polyhedron (3-D figure); what you get if you unfold a shape surface area the sum of the area of the faces of a 3D figure face a flat surface of a polyhedron (a 3D figure) edge the line segment along which two faces of a polyhedron intersect vertices a point where three or more edges intersect; the corners pyramid a polyhedron that has a polygon base and triangular lateral faces rectangular prism 3D figure where 6 faces are rectangles volume the number of cubic units needed to fill a given space
CCM6+ Unit 12 Surface Area and Volume page 3 MS 6 3-D Shapes/Polyhedrons WCPSS Video : http://bit.ly/6121ligon Polyhedron # of # of # of Net Faces Edges Vertices 1.) Triangular Prism Dimensions- Tri: b=6,h=5; Sqr: 6x6 Surface Area 2.) Rectangular Prism Dimensions- 6x6x10 3.) Square Pyramid Dimensions- Tri: b=6,h=5; Sqr: 6x6 4.) Triangular Pyramid Dimensions- Tri: b=6,h=5
CCM6+ Unit 12 Surface Area and Volume page 4 "I Can Identify Faces, Edges, and Vertices of 3D Figures." "I Can Represent a 3D Figure with a Net." 3-D Notes Face A surface of a 3D figure shaped like a polygon Edge The line segment which faces. Vertex A where or more edges intersect. Net: An arrangement of figures that can be to form a figure. Prisms Have identical bases. Named by the of their base. The sides of a prism are. Pyramids Have only base. Named by the of their base. The sides of a pyramid are. 3D Figure Net Properties of Figure Shape of Base: Name of Figure: Faces Vertices Edges Shape of Base: Name of Figure: Faces Vertices Edges Shape of Base: Name of Figure: Faces Vertices Edges
CCM6+ Unit 12 Surface Area and Volume page 5 3D Figure Net Properties of Figure Shape of Base: Name of Figure: Faces Vertices Edges Shape of Base: Name of Figure: Faces Vertices Edges Shape of Base: Name of Figure: Faces Vertices Edges Shape of Base: Name of Figure: Faces Vertices Edges Shape of Base: Real-World Examples of 3D Figures Name of Figure: Faces Vertices Edges Cube Triangular Prism Cylinder Sphere Cone Pyramid
CCM6+ Unit 12 Surface Area and Volume page 6 "I Can Identify Faces, Edges, and Vertices of 3D Figures." "I Can Represent a 3D Figure with a Net." Join each shape to the matching net. Annenberg Learner 3D figures and net: http://www.learner.org/interactives/geometry/3d_prisms.html
CCM6+ Unit 12 Surface Area and Volume page 7 Name the solid shape that can be formed by each net. 1. 2. 3. 4. 5. 6. Name each solid shape. How many faces, edges, vertices? 7. 8. 9. 10. 11. 12. 13. 14. 15.
Solids CCM6+ Unit 12 Surface Area and Volume page 8 About the Face Complete the Chart Number of Faces That Are Cube Rectangular Prism Triangular Prism Pentagonal Prism Hexagonal Prism Square Pyramid Triangular Pyramid Euler s Formula Complete the Chart Solid Number of Faces Numbers of Edges Number of vertices Cube Rectangular Prism Triangular Prism Pentagonal Prism Hexagonal Prism Square Pyramid Triangular Pyramid Pentagonal Pyramid Hexagonal Pyramid Euler s formula: The relationship exists between the faces, edges, and vertices of each solid is
CCM6+ Unit 12 Surface Area and Volume page 9 MS 6 Math Volume of Prisms WCPSS Video : http://bit.ly/6122ligon Volume is measured in units. 1. How many cubes are in the top layer? How many layers are there? How many total cubes would you need to build this structure? What is the: length? width? height? How could you use the dimensions to find the volume? How many cubes are in the top layer? How many layers are there? How many total cubes would you need to build this structure? What is the: length? width? height? How could you use the dimensions to find the volume? 2. Formula for VOLUME of a Rectangular Prism: V = Pause the video and try the problems on the back on your own! Then press play and check your answers with a color pen. 1. 2. 3. 4.
CCM6+ Unit 12 Surface Area and Volume page 10 "I Can Find the Volume of a Right Rectangular Prism by Applying the Appropriate Formula." EX 1. Find the Volume of the Rectangular Prism Ex 2. The area of the base of a rectangular prism is 12cm 2 and the height is 3 1 cm. Determine the 3 volume of the rectangular prism. Ex 3. A pet carrier company is creating a new size carrier in the shape of a rectangular prism. It has a width of 27 cm, a length of 7 cm, and a volume of 6,426 cubic cm. Find the height Ex 4. The volume of a rectangular sand box is 7 1 2 m3. The area of the base is 4 1 2 m2. What is the height of the sand box? Find the volume of any prism Ex 5. 1. Find the Area of the Base 2. Multiply the Area of the Base by the Height.
CCM6+ Unit 12 Surface Area and Volume page 11 Volume Formulas for Prisms Volume of a Cube Rectangular Prism Triangular Prism V = Bh V = Bh V = Bh V = s 3 V = (l w) h V = ( b h 2 Big B represents the area of the base polygon, and that base area is stacked h times. Instructions: Find the volume of each 3-D shape (prism): 1. 2. 3. 4. Find the volume of the truck s trailer. Challenge, how many of the boxes would fit inside the trailer?
CCM6+ Unit 12 Surface Area and Volume page 12 Volume Rectangular Prism Khan Academy Volume ; Volume of Triangular Prism Khan Academy Find the volume of each prism. 3. 5. 7. 8. 9. 10. 11. 12.
CCM6+ Unit 12 Surface Area and Volume page 13 VOLUME of RECTANGULAR PRISMS 1. A swimming pool with the dimensions shown below is filled two-thirds of the way full. It costs $0.04 per cubic ft. of water to fill the pool. How much did it cost to fill the pool two-thirds of the way full? 2. Jimmy s parents built him a sandbox in their back yard. Jimmy s parents purchased sand at the local home improvement store for $0.25 per cubic ft. How much would it cost to fill the sandbox? Assume the sandbox is filled to the top and leveled. 3. A cubic foot of salt water weighs approximately 64 pounds. If the fish tank with the dimensions shown below were filed with salt water. How much would the water weigh inside the fish tank? Assume that the fish tank is filled to the top 4. The Henderson family decided to construct a porch in their back yard made of concrete. The cost they were charged per cubic ft. was $3.33. The slab of concrete is 6 inches deep and the length and width of the top surface is 16 feet by 10 feet. How much did it cost the Henderson family to purchase the concrete necessary to construct their porch. 5. An Olympic-sized swimming pool has a volume of 88,263 cubic feet. An average adult can pump about 267 cubit ft. of blood per day.(this number varies depending on the person.) About how many days would it take based on the numbers given for a heart to pump enough blood to fill up an Olympic-sized pool? Round to the nearest day. 6. Examine the rectangular prism below. What is the difference in their volumes?
CCM6+ Unit 12 Surface Area and Volume page 14 MS 6 Math Volume with Fractional Units - Block Party http://bit.ly/6123ligon The cube shown to the right represents the unit cube. It has the dimensions 1 by 1 by 1. Color the faces of the unit cube blue. What is the volume of the cube? 1 x 1 x 1 = cubic unit The rectangular prism in Figure 1 is made up of some unit cubes as well as other cubes that have been cut in half. What are the dimensions of Figure 1? 2½ by by Color the faces of the unit cubes blue. Color the faces of the ½ cubes green. How many uncut (unit) cubes are in the figure? How many ½ cubes are in the figure? Without using the formula for finding volume, explain how you could find the volume of the prism. Now show how to find the volume using the formula. The rectangular prism in Figure 2 is made up of some unit cubes, some ½ cubes and some ¼ cubes. What are the dimensions of Figure 2? by by 2½ Color the faces of the unit cubes blue. Color the faces of the ½ cubes green. Color the faces of the ¼ cubes red. How many unit cubes are in the figure? How many ½ cubes are in the figure? How many ¼ cubes are in the figure?
CCM6+ Unit 12 Surface Area and Volume page 15 Without using the formula for finding volume, explain how you could find the volume of the prism. Now show how to find the volume using the formula. The rectangular prism in Figure 3 is made up of unit cubes, ½ cubes, ¼ cubes and 1/8 cubes. What are the dimensions of Figure 3? by by 4½ Color the faces of the unit cubes blue. Color the faces of the ½ cubes green. Color the faces of the ¼ cubes red. Color the faces of the 1/8 cube yellow. How many unit cubes are in the figure? many ½ cubes are in the figure? How How many ¼ cubes are in the figure? How many 1/8 cubes are in the figure? Without using the formula for finding volume, explain how you could find the volume of the prism. Now show how to find the volume using the formula. A rectangular prism with the dimensions of 2 by 3 by 4 has a volume of 24. Name at least 3 other rectangular prisms (length, width, and height) with at least one fractional dimension that have a volume of 24.
CCM6+ Unit 12 Surface Area and Volume page 16 MS 6 Math Volume Fractional Units WCPSS Video Notes : http://bit.ly/6124ligon Find the volume of these rectangular prisms with fractional unit measurements. 1 st Step: Change the measure of each dimension to a fractional measurement (improper fraction.) 2 nd Step: Diagram the fractional units on each of the prisms (to the best of your ability.) 3 rd Step: Multiply the three fractional units together, each measure representing a dimension. 4 th Step: How many whole cubic units and fractional cubic units will each of these prisms hold?
CCM6+ Unit 12 Surface Area and Volume page 17 Volume of Rectangular Prism Volume with fractional Units 1. If the edge of each smaller cube which comprises the larger rectangular prism is of an inch in length, what is the volume of the entire rectangular prism? 2. If a young child builds the structure below using blocks that are 1 cubic inch in size, how many blocks would be needed to complete the structure? 3. How many cubes with an edge length of 1/3 of an inch would it take to make a cube with a volume of one cubic inch? 4. A pool has a length and width of 12 by 15 feet and is 10 feet deep. How long will it take to fill the entire pool if it is being filled at a rate of one-eighth of a cubic foot per second? Express your answer in hours. 5. A right rectangular prism has edges of, 2 1 in, 2 in 4 and 1 1 in. How many cubes with lengths of 1 in would 2 4 be needed to fill the prism? What is the volume? 6. Find the volume of a rectangular prism with dimensions 1 1 in, 1 1 in and 2 1 in.how many cubes 2 2 2 with lengths of 1 in would be needed to fill the 2 prism?
CCM6+ Unit 12 Surface Area and Volume page 18 7. A follower box is 3 feet long, 1 3 4 feet wide, and 1 2 feet deep. How many cubic feet of dirt can it hold? 8. Mr. White is trying to store boxes in a storage room with length of 18ft, width of 10ft and height of 8ft. How many boxes can fit in this space if each is box is 2 1 feet long 4 11 feet wide and 1 feet deep? 2 9. Alexia s bathroom has a tub in the shape of a rectangular prism with a length of 1.5 meters, a width of 0.5 meter, and a height of 0.4 meter. How many cubic feet of water can it hold? 1 10. A right rectangular prism has edges of 1, 1 4 1 1 and 1. How many cubes with side lengths of 2 4 would be needed to fill the prism? What is the volume of the prism? 11. Michael is building a sand box that is 6 feet wide, 3 feet long, and 15 inches high. How many cubic feet of sand with the box hold? 12. A box with the dimensions 5 x 8 x 12 is filled up two-thirds of the way. How much volume of the box is filled up? 13. Mary s swimming pool is shaped like a rectangular prism. The volume of the pool is 12,500 cubic feet. If the base of the pool is 25 feet by 50 feet, what is the depth of the pool? 14. How many cubic feet of concrete will be needed for a driveway that will be 16 feet long, 10 feet wide and ½ of a foot thick?
CCM6+ Unit 12 Surface Area and Volume page 19 Volume of Composite figures. Find the volume of each composite figures. 1. 2. 3. 4..
CCM6+ Unit 12 Surface Area and Volume page 20 MS 6 Math Surface Area & Volume WCPSS Video : http://bit.ly/6125ligon For all problems: Find the amount (surface area) of glass needed for each aquarium, as well as the capacity (volume), in gallons, that each aquarium will hold. Don t forget that the lid (top base) will need glass as well. (Hint: 231 in 3 = 1 gallon, first find the volume in cubic inches, then convert to gallons.) 1.) 18 in. 24 in. 12 in. 3.) Students in CCM6 and CCM6 Plus do not have to know the formula for volume of a triangular prism, but should be able to use a formula when given. V = Bh or V = ( bh 2 ) h
CCM6+ Unit 12 Surface Area and Volume page 21 MS 6 Math Surface Area & Nets WCPSS Video http://bit.ly/6126ligon 1 st Step: Diagram a net for each 3-D object. Label the dimensions of each polygon (face.) 2 nd Step: Find the area of each face. 3 rd Step: Total the areas of each face to find the total surface area. 1.) 3-D Object Net Diagram Total Surface Area 2.) 3.) 4.)
CCM6+ Unit 12 Surface Area and Volume page 22 Surface Area I Can Find the Surface Area of a 3D Figure by Finding the Area of Each of its Faces With and Without the Use of a Net. Example 1 Find the Surface Area of the Triangular Prism Left Face Back Face Right Face Top Base Bottom Base Total Surface Area: Example 2) Find the Surface Area of the Rectangular Pyramid. Left Face Back Face Right Face Front Face Bottom Base Total Surface Area:
CCM6+ Unit 12 Surface Area and Volume page 23 Example 3) Find the Surface Area of the Rectangular Prism. Left Face Front Face Top Base Total Surface Area: Right Face Back Face Bottom Base Example 4) Find the Surface Area of the Regular Triangular Pyramid. Left Face Front Face Right Face Total Surface Area: Total Surface Area:
CCM6+ Unit 12 Surface Area and Volume page 24 For each shape, label the 3D shape. Then Draw /label it s net. Finally, find its total surface area. 1. A square pyramid with base length 15 cm and slant height 22 cm 2. Fatima is wrapping a gift box for her nephew s birthday. The box s dimensions are 16 inches long by 10 inches wide by 5 inches high. What is the surface area of the box? 3. A rectangular prism with dimensions 2 1 in, 4. An equilateral triangular prism with triangle side 4 4 3 in, and 4 11in. length 3 ft, triangle height 2.5 ft, and prism height 9 2 ft. 5. Cinema theaters created a new popcorn box in the shape of a rectangular prism. The new popcorn box has a length of 6 inches, a width of 3. 5in, and a height of 3. 5in but does not include a lid. How much material is needed to create the box? 6. Max is mailing 2 packages to his friend in New York. He measured the boxes and found the following dimensions: Box 1 Box 2 Length: 17 in Length: 17 in. Width: 11 in. Width: 11 in. Height: 4 in. Height: 8 in. He needs to cover the boxes with paper. He estimates that it will take about twice as much paper to cover Box 2 since it is twice as tall. Is he correct? Explain and justify your answer. 7. Kyle is painting the front door of his house. The dimensions of the door are 80 inches by 36 inches by 2 inches. If he paints all of the surfaces, how much area will he paint? Explain. 8. A cube with side length 5 1 3 cm.
CCM6+ Unit 12 Surface Area and Volume page 25
CCM6+ Unit 12 Surface Area and Volume page 26 Surface Area Practice A. Draw and label the net for each figure. B. Find the surface area of each figure. 1. 2. 3. 4. 5. 6. 7. 8. 9. Challenge: You plan to build a birdhouse with one square doorway as shown. How many square centimeters of wood do you need to make the birdhouse?
CCM6+ Unit 12 Surface Area and Volume page 27 Surface Area Review Homework Sheet Match the 3D shape to the net. 5. Which net cannot be used for this shape? 6. Which net can be used for this shape? 7. Which net can be folded to make a 3D shape?
CCM6+ Unit 12 Surface Area and Volume page 28 Find the total surface area of the shape. Nets are provided to help you.
CCM6+ Unit 12 Surface Area and Volume page 29 Unit 12 Pre-test : Surface Area and Volume (calculator active) 1. An aquarium has a volume of 9,525 cm 3. The aquarium is 12.5 centimeters wide and 30 centimeters tall. How long is the aquarium? 2. The model shows a rectangular prism with 3 2 5 2 dimensions,, and inches. What is the volume of one of the small cubes? 5 2 3.Find the volume of the rectangular prism given the area of the base is 316 m 2 and the height is 58 m. 4. A right rectangular prism has edges of 1 1 1, 1 and 1. 4 2 a. What is the volume of the prism? b. How many cubes with side lengths of would be needed to fill the prism? 1 4 5. Find the volume of the figure below; each cube has an edge length of ½ inch. 6. Draw a net for the three-dimensional figure below.
CCM6+ Unit 12 Surface Area and Volume page 30 7. Draw a net and find the surface area of the figure below. 8. Elaine s room is in the shape of a rectangular prism 15 feet long, 12 feet wide and 10 feet tall. Elaine paints the four walls and the ceiling but not the floor. How much surface area does Elaine paint? 9. What three-dimensional figure does the net represent? How many faces, vertices and edges does the three-dimensional figure have? 10. Using the net what is the surface area of the figure below? Name of three-dimensional figure Faces - Vertices - Edges -
CCM6+ Unit 12 Surface Area and Volume page 31 11. Find the volume of the rectangular prism below. How many times larger would the volume of this figure be compared to the volume of the prism made up of cubes of side length 1 2 unit? How many times larger would the volume of this figure be compared to the volume of the prism made up of cubes of side length 1 4 unit? Use your findings to explain what would happen to the volume of the same figure if you packed the prism with cubes of side length 1 5 unit. 12. Nets can be made up of lots of different shapes. Which 3-dimensional shapes have nets that include some squares? Rectangles? Triangles? List at least two 3-dimensional shapes in each column. These 3-D shapes have at least one SQUARE in their nets These 3-D shapes have at least one RECTANGLE in their nets These 3-D shapes have at least one TRIANGLE in their nets Describe how the nets for a rectangular prism with a lid and the same prism with no lid would look different.