Three-Dimensional Figures and Nets

Transcription

1 Lesson 11.1 Reteach Three-Dimensional Figures and Nets Solid figures have three dimensions length, width, and height. They can be named by the shapes of their bases, the number of bases, and the shapes of their lateral faces. Identify and draw a net for the solid figure. Step 1 Describe the base of the figure. The base is a square. Step Describe the lateral faces. The lateral faces are triangles. So, the figure is a square pyramid. Step 3 Name the shapes to be used in the net. Then make a sketch. Draw a square for the base, and four triangles for the lateral faces. Identify and draw a net for the solid figure. 1.. figure: figure: 11-5 Reteach

2 Lesson 11.1 Enrich Solid Identification Use the figures below to answer the questions. B A C D 1. What is the name of Figure A?. What is the name of Figure B? How many faces, edges, vertices, and bases does the figure have? How many faces, edges, vertices, and bases does the figure have? 3. What is the name of Figure C? 4. What is the name of Figure D? How many faces, edges, vertices, and bases does the figure have? How many faces, edges, vertices, and bases does the figure have? 5. Stretch Your Thinking Draw a triangular pyramid in the rectangle at the top of the page. How many faces, edges, vertices, and bases does the figure have? 6. Describe the similarities and differences between a triangular prism and a rectangular prism Enrich

3 Lesson 11. Reteach Explore Surface Area Using Nets The net of a solid figure shows you all of the faces or surfaces of the figure. A net can help you find the surface area of a figure. Find the surface area of the rectangular prism. 1 in. 4 in. in. Step 1 Make a net of the rectangular prism. The prism has 6 rectangular faces, so the net has 6 rectangles. E A C B D F Step Find the area of each face of the prism. First Way: Count the unit squares on each rectangle to find its area. Second Way: Calculate the area of each rectangle by multiplying length 3 width. A: 8 unit squares B: 8 unit squares C: 4 unit squares D: 4 unit squares E: unit squares F: unit squares Step 3 Add the areas of all the rectangular faces. 8 unit squares 8 square inches So, the surface area of the rectangular prism is 8 square inches (in. ). Use the net to find the surface area of the prism. 1. a. Find the area of each face. 3 cm 4 cm A: B: 5 cm C: D: A C E D B F E: F: b. Add: A 1 B 1 C 1 D 1 E 1 F 5 c. The surface area is cm Reteach

4 Lesson 11. Enrich Nets, Nets, Nets Draw and label two nets for the rectangular prism. Then use the nets to find the surface area. 1. length 9 cm width 3 cm height 5 cm Surface area:. length 4 in. width 6 in. height in. Surface area: 3. length 7.3 m width 3. m height 8.5 m Surface area: 11-8 Enrich

5 Algebra Surface Area of Prisms You can find the surface area of a figure by adding the lateral surface area to the sum of the areas of the bases. 1 in. Lesson 11.3 Reteach 8 in. 6 in. 10 in. Use a net to find the surface area. Step 1 Draw a net. Note any faces that have equal areas. A 6 in. 10 in. 8 in. 6 in. 1 in. D C B 1 in. Step Both triangular bases have the same area. 6 in. 1_ Base A: A 5 bh = 1_ in. Base E: A 5 4 in. E Step 3 Find the areas of the rectangular faces. Face B: A 5 lw in. Face C: A 5 lw in. Face D: A 5 lw in. Step 4 Add the areas: A 1 B 1 C 1 D 1 E in. So, the surface area of the triangular prism is 336 square inches (in. ). Use a net to find the surface area in. 8 in.. 13 cm 16 in. 8 cm 17 in. 6 cm 3. m 11 m 15 m 4. 8 ft 8 ft 8 ft 11-9 Reteach

6 Lesson 11.3 Enrich Use Surface Area to Make Food Packages You ve been hired to design packaging for the following food items. Match each box with its description. Then find the surface area of the indicated material. Item: Green Energy Tea Bags The tea is sold in a cube-shaped box with sides that are 9.5 in. long. The box is fully lined with tissue paper. Box: Item: Country Cabin Corn The box has length 14.5 in., width 9. in., and height 5 in. The box is wrapped in two layers of freezer paper. Box: Tissue paper area: Item: Tasty Flakes and Raisins Cereal The top and bottom of the cereal box is 10 in. by.5 in. The box is 1.3 in. high. The cardboard is coated with dye, except on the bottom. Box: Total paper area: Item: V.I.P. Select Chocolates Two boxes are stacked and tied with a ribbon. Each box is fully wrapped in decorative paper. Bottom box: 8 in. 3 6 in. 3 3 in. Top box: 5 in. 3 3 in. 3 in. Box: Dyed area: Total paper area: A C E B D 1. Stretch Your Thinking Tissue paper for the tea boxes costs $0.01 per square inch. How much would it cost to line a dozen boxes, to the nearest penny?. Green Tea decided they do not want tissue paper on the top of the box. Using the surface area you found, explain how to find the new surface area Enrich

7 Lesson 11.4 Reteach Algebra Surface Area of Pyramids To find the surface area of a pyramid, add the area of the base to the lateral area. The lateral area is the combined area of the triangular faces. Find the surface area of the square pyramid. Step 1 The base is a square with side length of 6 in. Use the formula A 5 s to find the area. Substitute 6 for the variable s. 8 in. A in. 6 in. Step The lateral faces are four triangles with base of 6 in. and height of 8 in. Find the area of one triangular lateral face using the formula A 5 1_ bh. Substitute 6 for b and 8 for 1_ h. A 5 (6)(8) 5 4 in. 6 in. 6 in. 8 in. Step 3 Multiply by 4 to find the total lateral area. L in. Step 4 Add the area of the base and the lateral area. S 5 36 in in in. So, the surface area of the square pyramid is 13 square inches (in. ). Use a net to find the surface area of the square pyramid cm 4 ft 9 cm a. Area of the base: b. Area of one triangular lateral face: 10 ft a. Area of the base: b. Area of one triangular lateral face: c. Total lateral area: d. Total surface area: c. Total lateral area: d. Total surface area: Reteach

8 Lesson 11.4 Enrich Surface Formulas Draw and label a net for the solid figure. Then use the net to write an algebraic formula for the surface area. Apply your formula using the given values. 1. Rectangular prism Net h l w ( ) 1 ( ) 1 ( ) Evaluate for l 5 5 in., w 5 3 in., h 5 4 in.. Triangular pyramid; faces and base have the same measurements Net h b Evaluate for b 5 6 ft and h 5 5. ft. 3. How would the formula for surface area of a triangular pyramid be different if the base and other faces were not equivalent? 11-1 Enrich

9 Lesson 11.5 Reteach Fractions and Volume Find the volume of a rectangular prism that is 1_ units long, units wide, and 11_ units high. 1 1 units 1 units units Step 1 Stack cubes with 1_ -unit side length to form a rectangular prism. 1_ 1_ 1_ 1_ 1_ 1_ Length: 5 cubes units 1_ 1_ 1_ 1_ Width: 4 cubes units Height: 3 cubes 5 1_ 1 1_ 1 1_ 5 1 1_ units Step Count the total number of cubes. 60 cubes Step 3 It takes 8 cubes with 1_ -unit side lengths to make 1 unit cube. So, each smaller cube has 1_ 8 the volume of a unit cube. Divide 60 by 8 to find how many unit cubes it would take to form the prism. Write the remainder as a fraction and simplify. 4_ 1_ is 7 R So, the volume of the prism is 7 1_ cubic units. 1. Find the volume of the rectangular prism. units 1 1 units 1 1 units a. Stack cubes with 1_ -unit side lengths to form the prism. b. Count the cubes. c. Divide by is R d. The prism has a volume of cubic units Reteach

10 Lesson 11.5 Enrich Packing Boxes Sometimes products are less expensive when bought in larger packages. Have you ever wondered why? A B 1. Box A is a cube with a side length of 1 1 inches. a. Find the surface area of the box.. Box B is a cube with a side length of 1 inch. a. Find the surface area of the box. b. Find the volume of the box. b. Find the volume of the box. 3. How many of Box B would it take to fill Box A? 4. Find the total surface area for 10 Box Bs. 5. Which has a smaller surface area, Box A, or 10 Box Bs? 6. The company pays for boxes according to how many square inches of cardboard each box uses. Which is less expensive to use, Box A or 10 Box Bs? How do you know? Enrich

11 Lesson 11.6 Reteach Algebra Volume of Rectangular Prisms You can find the volume of a prism by using the formula V 5 Bh. V stands for volume, B stands for the area of the base, and h stands for the height. For a rectangular prism, any face can be the base, since all faces are rectangles. 3 1_ ft 5 ft 1_ ft 1 yd 1 yd 1 yd Find the volume of the rectangular prism. Step 1 Find the area of the base. Step Multiply the area of the base by The base is 1_ ft by 31_ ft. the height. A 5 l 3 w V 5 Bh 1_ A 5 ft 3 31_ ft 5 83_ 4 ft 3_ 3_ V ft So, the volume of the rectangular prism is 43 3_ 4 ft 3. Find the volume of the cube. Step 1 Because the length, width, and height are all equal, you can use a special formula. V 5 Bh 5 l 3 w 3 h V 5 s 3 So, the volume of the cube is 15 5_ 8 yd 3. 1_ Step Substitute for s. V 5 s 3 5 ( 1_ ) 3 5 ( 5_ ) 3 V 5 5_ yd 3 5_ _ 8 yd3 yd yd 5 8 yd3 Find the volume m. 11 in. 5 m 4 m 11 in. 11 in _ cm 4 1_ cm cm _ ft 30 ft 16 ft Reteach

12 Lesson 11.6 Enrich Aquarium Volumes on Display A pet store is selling aquariums. You have been asked to find the volumes of the aquariums in cubic inches. After finding the volumes, answer the questions that follow. 15-Gallon Aquarium 13 in. $35 5-Gallon Aquarium $ in. 4 in. 1 in. 4 in. 1.5 in. Volume: Volume: 55-Gallon Aquarium $ Gallon Aquarium $175 1 in in. 3 in. 8.8 in. 8 in. 46 in. Volume: Volume: 1. A customer says he has a rectangular aquarium with a length of 37 in., a width of 19 in., and a height of 19 in. What is the volume of the aquarium?. The gallon sizes above are approximate. To which size is the customer s aquarium closest? 3. Stretch Your Thinking What is the volume of the 5-gallon aquarium in cubic feet? Round to the nearest tenth. 4. Which of the four aquariums has the best value, based on the advertised gallon sizes? Explain your answer Enrich

13 Problem Solving Geometric Measurements Leslie stores gardening supplies in this shed shaped like a rectangular prism. What is the area of the ground covered by the shed? 1 feet Lesson 11.7 Reteach 8 feet 7 feet Read the Problem What do I need to find? the area of the ground covered by the shed I need to find. What information do I need to use? I need to use the dimensions of. the shed How will I use the information? which measure First, I will decide to. use Then I will choose a formula replace the variables and evaluate the formula I can use to calculate this measure. Finally, I will. Solve the Problem Choose the measure area, surface area, or volume that gives the area of the ground covered by the shed. Explain. area; The area covered by the shed is the same as the area of the. base of the shed Choose an appropriate formula. Replace the variables l and w in the area formula with their values in the dimensions of the shed. 1 l 5 ft Evaluate the formula A ft A 5 l 3 w 8 w 5 ft 8 Solve. 1. Leslie is covering bricks with paint. Each brick is 8 in. long, 4 in. wide, and in. high. How many square inches will Leslie paint on each brick?. Leslie s planting box is shaped like a rectangular prism. It is 60 cm long, 35 cm wide, and 40 cm high. How many cubic cm of soil will Leslie need to fill the box? Reteach

14 Lesson 11.7 Enrich Volume at the Aquarium The County Aquarium has some new exhibits. The Shark Tank is a closed aquarium that has a length of 73 inches, a width of 40 inches, and a height of 144 inches. The glass is 6 inches thick on all sides. 1. Draw and label the tank.. Use a formula to find the inside volume of the tank in cubic feet. Formula: 3. Each cubic foot will hold approximately 7.48 gallons of water. To the nearest gallon, how many gallons of water will the tank hold? Volume: 4. Each gallon of water is approximately 3.8 liters. Predict whether the number of liters will be greater than or less than the number of gallons. Explain your reasoning. 5. To the nearest liter, how many liters of water does the shark tank hold? 6. Explain why you can or cannot figure out the shape of a tank if you only know how many gallons it holds Enrich