# CHAPTER 11 - mrjuarezclass

embed*EMATH8_C11_v3b 8/31/05 9:20 AM Page 369 CHAPTER 11 Geometry and Measurement Relationships GOALS You will be able to • develop and apply formulas to calculate the surface area and volume of a cylinder • solve problems that involve the surface area and volume of a cylinder • investigate the properties of Platonic solids • determine how the number of faces, edges, and vertices of a polyhedron are related (Euler’s formula) *EMATH8_C11_v3b 8/31/05 9:21 AM Page 370 Getting Started You will need • a calculator • centimetre grid paper Designing a Juice Container Hoshi and Tran are designing a container to hold 1 L of juice. They decide to try the shape of a triangular prism. dimensions can Hoshi and Tran use for the juice ? What container? A. What is the least possible volume, in cubic centimetres, for the container? (1 cm3 1 mL) B. About 10% of the space in the container will be empty. How many cubic centimetres do Hoshi and Tran need for the total volume of the container? C. If the container is 10 cm deep, what must the area of the triangular base be? D. What dimensions could Hoshi and Tran use for the triangular base? 370 Chapter 11 10 cm NEL *EMATH8_C11_v3b 8/31/05 9:21 AM Page 371 Do You Remember? 5. Calculate the volume of each prism. a) 1. What unit (centimetres, square centimetres, cubic centimetres, litres, or millilitres) would you use to measure each of the following? a) b) c) d) the quantity of fuel in a car’s gas tank the length of trim on a blanket the amount of paint to cover a playhouse the space inside a storage box 2. Draw a net for each prism using centimetre grid paper. Calculate the surface area of each prism to the nearest square centimetre. a) 12 cm 4.5 cm 6.2 cm b) 3 cm 4 cm 6 cm 6. How much water would you need to fill each prism in question 5? (1 cm3 1 mL) 7. List the number of edges, faces, and vertices on each prism. b) 4 cm 4 cm 8.4 cm a) 5 cm 5 cm 4 cm 3. Sarah is covering the faces of this wooden box with fabric for a gift. How much fabric will she need to cover the box? 6 cm 3 cm 15 cm 4. Calculate the area and circumference of each circle. b) 8. Centimetre cubes need to be packed in a box that has a volume of 36 cm3. Sketch three possible boxes that are rectangular prisms. 9. Think of this cement traffic barrier as a rectangular prism with a trapezoidal prism on top. 23.0 cm 80.0 cm a) 25.5 cm 4.0 cm 66.0 cm b) 4.5 m NEL 380.0 cm a) What is the area of the base of each prism? b) What is the volume of each prism? c) What is the total volume of the cement needed to make the traffic barrier? Geometry and Measurement Relationships 371 *EMATH8_C11_v3b 8/31/05 9:21 AM Page 372 You will need • paper • scissors • a calculator 11.1 Exploring Cylinders GOAL Explore the relationship between the dimensions of a cylinder and the dimensions of its net. Explore the Math Maria and Benjamin are playing a game at their school math fair. They are shown a lid for a cylindrical container and four labels. To win a prize, they have to choose the label that matches the lid. They can ignore any overlap on the labels. 4.2 cm 13.3 cm 35.7 cm 10.1 cm Label A 26.4 cm 10.7 cm Label B 13.4 cm 10.7 cm 27.3 cm Label C can you use the size of a lid to predict the width of ? How the label? A. Cut out paper models of the lid and the rectangular labels. B. Use the rectangles to construct cylinders without bases. C. Find the cylinder that matches the lid. Make a net for this cylinder. D. Estimate the radius, circumference, and area of the lid. E. Which measurement of the lid—the radius, circumference, or area— does the width of the label match? F. Compare the widths of the labels with the measurement of the lid you selected in step E. Which label goes with the lid? Label D Communication Tip The curved side of a cylinder has height and width. When the curved side is unrolled into a rectangle, the height of the cylinder becomes the length of the rectangle. width height Reflecting 1. How did you know which label went with the lid? 2. How would the labels for two cylinders with the same height but different diameters compare? width length 3. If you knew the height and radius of a cylinder, how would that help you draw its net? 372 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:22 AM Page 373 NETS OF CONES The net for a cone provides clues that tell you how flat the cone will be when it is constructed. Cone A Cone B Cone C Net X Net Y Net Z A. Draw three circles, each with a radius of 8 cm. Cut out your circles. B. Cut out a section of one circle, along two radii of the circle, to create a net for a bottomless cone. Measure the angle. You will need • • • • • a compass scissors a ruler a protractor tape C. Tape together the cut edges to make the cone. D. Repeat steps B and C to make two different-sized nets and cones. E. What relationship do you notice between the angles of your nets and the flatness of your cones? F. Which net (X, Y, or Z) goes with which cone (A, B, or C)? NEL Geometry and Measurement Relationships 373 *EMATH8_C11_v3b 8/31/05 9:22 AM Page 374 You will need • a calculator • centimetre grid paper • a ruler • a compass 11.2 Surface Area of a Cylinder GOAL Develop and apply a formula for calculating the surface area of a cylinder. Learn about the Math 3.5 cm Toma and Maria are wrapping tea lights to sell for a school fundraiser. They are wrapping the tea lights in stacks of five. 1.5 cm much paper do Toma and Maria ? How need to wrap each stack of tea lights? Example 1: Estimating surface area with grid paper Use a net to estimate the amount of wrapping paper that Toma and Maria need to wrap each stack of five tea lights. Toma’s Solution I imagined unwrapping the package and laying the paper flat. I traced the circular base and top of the cylinder on grid paper. curved side 3.5 cm top 3.5 cm 7.5 cm 1.5 cm 3.5 cm base 11 cm The height of each tea light is 1.5 cm, so the height of the curved side is 5 1.5 cm 7.5 cm. The width of the curved side is the same as the circumference of the base. The diameter of the base is 3.5 cm, so the circumference is 3.5 cm 11 cm. I counted the squares in the net. There are about 10 squares in each circle. There are about 7.5 11 83 squares in the curved side (rectangle). Each square has an area of 1 cm2. Total surface area area of base area of top area of curved side 10 cm2 10 cm2 83 cm2 103 cm2 Toma and Maria need about 103 cm2 of wrapping paper for each stack of tea lights. 374 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:22 AM Page 375 Example 2: Calculating surface area using a formula Calculate the amount of wrapping paper that Toma and Maria need to wrap each stack of five tea lights. Maria’s Solution The base and top of each stack are congruent faces, so they have the same area. I just need to determine the area of one face, and then I can double it. The diameter is 3.5 cm, so the radius is 3.5 cm 2 1.8 cm. base Area of top and base 2 r 2 2 (1.8 cm)2 top 3.5 cm The area of the top and base is 20.4 cm2. I rounded to one decimal place because this is how the radius is given in the problem. If I laid out the curved side of the package, it would form a rectangle. Its width would be equal to the circumference of the base. Its length would be equal to the height of the package. So, its area is equal to the circumference of the base multiplied by the height of the package. 3.5 cm Area of curved surface circumference of base height d height ( 3.5 cm) 7.5 cm 11.0 cm 7.5 cm 7.5 cm The area of the curved surface is 82.5 cm2. Total surface area area of top and base area of curved surface 20.4 cm2 82.5 cm2 102.9 cm2 Toma and Maria need 102.9 cm2 of wrapping paper for each stack of tea lights. Reflecting 1. What part of the net for a cylinder is affected by the height of the cylinder? 2. What part of the net for a cylinder is affected by the size of the base? 3. Write a formula to calculate the surface area of a cylinder. NEL Geometry and Measurement Relationships 375 *EMATH8_C11_v3b 8/31/05 9:22 AM Page 376 Work with the Math Example 3: Calculating surface area using a formula 2.0 cm Calculate the surface area of this cylinder. 3.0 cm Solution Surface area of cylinder area of base and top area of curved surface Sketch the faces, and calculate the area of each face. 2.0 cm Area of base and top 2 area of base 2 r 2 2 3.14 (2.0 cm)2 25.1 cm2 3.0 cm 12.6 cm Width of rectangle circumference of circle 2r 2 3.14 2.0 cm 12.6 cm Area of curved surface area of rectangle length width 3.0 cm 12.6 cm 37.8 cm2 Surface area of cylinder area of base and top area of curved surface 25.1 cm2 37.8 cm2 62.9 cm2 The surface area is 62.9 cm2. A Checking B 4. Calculate the surface area of each cylinder. a) 5 cm b) 3.6 cm Practising 5. Use each net to determine the surface area of the cylinder. 6 cm a) b) 4.5 cm 8.0 cm 4 cm 5 cm 14.5 cm 376 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:22 AM Page 377 6. Three cylinders have bases that are the same size. The area of the base is 10.0 cm2. Determine the surface area of each cylinder, given its height. a) 8.0 cm b) 6.5 cm c) 9.4 cm 7. Calculate the surface area of each cylinder. 2.1 m a) c) 2.5 cm 8.3 m 10.3 cm b) 23.0 m 10. Explain why two cylinders that are the same height can have different surface areas. 11. “Anik” is the name of a series of Canadian communications satellites. The first Anik, shown here, was launched in 1972. It was a cylindrical shape, 3.5 m high and 190 cm in diameter. All satellites are wrapped with insulation because the instruments inside will not work if they become too cold or hot. What was the approximate area of the insulation used to wrap the first Anik? 2300 cm 8. Describe how you would determine the surface area of a potato-chip container that is shaped like a cylinder. 9. a) This railway car is 3.2 m in diameter and 17.2 m long. Calculate its surface area. 12. A games shop sells marbles in clear plastic cylinders. Four marbles fit across the diameter of the cylinder, and 10 marbles fit from the base to the top of the cylinder. Each marble has a diameter of 2 cm. What is the area of the plastic that is needed to make one cylinder? C b) Estimate the cost of painting the outside of the railway car, if a can of paint covers an area of 40 m2 and costs $35. NEL Extending 13. Gurjit has a CD case that is a cylindrical shape. It has a surface area of 603 cm2 and a height of 10 cm. What is the area of the circular lid of the CD case? Geometry and Measurement Relationships 377 *EMATH8_C11_v3b 8/31/05 9:22 AM Page 378 You will need 11.3 Volume of a Cylinder GOAL Develop and apply a formula for calculating the volume of a cylinder. • centimetre grid paper • a compass • centimetre linking cubes • a calculator Learn about the Math 2.0 cm 4.0 cm 8.0 cm 6.0 cm 4.0 cm 2.0 cm Benjamin says, “The jar with the greatest volume holds the most jellybeans. One jellybean has a volume of about 1 cm3. If I can calculate the volume of each jar, I can estimate the number of jellybeans it holds. I’ll use models of the jars to estimate their volumes. I know that the bottom of each jar is a circle. I’ll estimate the number of centimetre cubes that will cover the bottom. I’ll stack centimetre cubes to determine the number of layers.” ? Which jar holds the most jellybeans? A. On centimetre grid paper, draw a circle with a radius of 6.0 cm to model the base of the first jar. Estimate the area of the base. 6.0 cm B. Stack centimetre cubes to model the height of the jar. How many layers of cubes will fit inside the jar? C. Estimate the volume of the first jar. D. Repeat steps A to C for the other two jars. E. Which jar holds the most jellybeans? Explain your answer. 378 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:22 AM Page 379 Reflecting 1. Why do both the radius of a cylinder and its height matter when you are estimating its volume? 2. How does Benjamin’s strategy show how to use the area of the base of a cylinder to determine the volume of the cylinder? 3. Write a formula for calculating the volume of a cylinder. Work with the Math Example 1: Calculating volume using radius Calculate the volume of this cylinder. 5.0 cm 6.0 cm Hoshi’s Solution Volume of cylinder area of base height r 2 height 3.14 (5.0 cm)2 6.0 cm 471.0 cm3 The volume of a cylinder is calculated like the volume of a prism: area of base height. I calculated the volume using the formula and 3.14 for . The volume of the cylinder is 471.0 cm3. Example 2: Estimating volume using diameter Estimate which cylinder has the greater volume. 10 cm 14 cm 14 cm A B 10 cm Chad’s Solution Cylinder A The diameter of cylinder A is 10.0 cm, so the radius is 10.0 2 5.0 cm. For an estimate, I can use rounded measurements for easier mental math. Volume of cylinder A area of base height r 2 height (5 cm)2 14 cm 25 cm2 16 cm 400 cm3 Cylinder B has the greater volume. NEL Cylinder B The diameter of cylinder B is 14.0 cm, so the radius is 7.0 cm. Volume of cylinder B area of base height r 2 height (7 cm)2 10 cm 50 10 cm 500 cm3 Geometry and Measurement Relationships 379 *EMATH8_C11_v3b A 8/31/05 9:22 AM Page 380 Checking 4. Estimate the volume of each cylinder. 5m a) 7. a) Determine the volume of Mandy’s mug. 10.0 cm b) About how many millilitres of liquid will it hold? 4m 20.0 cm 3.2 cm b) 8. Calculate the volume of each cylinder. 5 cm a) 10.5 cm 8 cm 5. Calculate the volume of each cylinder. 14.4 cm a) b) 12.6 cm b) 3.1 m 4.1 m 7.3 cm 9. Cosmo’s family has a pool like this. 2.5 cm 5.4 m 120 cm B Practising 6. Estimate the volume of each cylinder. a) 4 cm 8 cm b) 10.2 cm 5.8 cm 380 Chapter 11 a) What is the volume of the pool? b) How many litres of water will the pool hold? c) How long will it take to fill the pool at a rate of 50 L/min? 10. Tennis balls are sold in cylinders. Each cylinder has a height of about 22 cm and a diameter of about 7 cm. Estimate the volume of the cylinder. NEL *EMATH8_C11_v3b 8/31/05 9:22 AM Page 381 8 cm 11. A cylindrical candle is sold in a gift box that is a squarebased prism. What is the volume of empty space in 16 cm the box? 16. Which container holds more, the cylinder or the triangular prism? Justify your answer. 2.1 cm 6.0 cm 3.5 cm 10.0 cm 12. Copy and complete this chart for cylinders. 4.0 cm Radius of base Diameter of base Height 12.0 m 11 cm 5.0 m 4 cm Volume 307.7 cm3 3.5 cm 2.0 m 17. Which holds more, a cylinder with a height of 10.0 cm and a diameter of 7.0 cm or a cylinder with a height of 7.0 cm and a diameter of 10.0 cm? Explain your answer. 226.1 m3 C 13. The area of the base of a cylinder is 50.2 cm2. The volume of the cylinder is 502.4 cm3. Determine the height of the cylinder. 14. A lipstick tube has a volume of 25.1 cm3 and a diameter of 2.0 cm. What is the height of the tube? 15. The volume of each cylinder below is 0.3040 m3. Solve for the unknown measure. 4.0 cm a) ■ cm ■ cm b) 8.8 cm NEL Extending 18. These two metal cans both hold the same quantity of soup. 10.0 cm 7.5 cm 8.0 cm a) What is the height of the can of mushroom soup? Show your solution. b) Which can uses more metal? Explain. 19. Suppose that the radius of a cylinder is the same as its height. What would happen to the volume of the cylinder if its radius were doubled and the height stayed the same? 20. These two containers each hold 1 L of liquid. What might their dimensions be? Geometry and Measurement Relationships 381 *EMATH8_C11_v3b 8/31/05 9:22 AM Page 382 You will need • a calculator 11.4 Solve Problems Using Diagrams GOAL Use diagrams to solve measurement problems. Learn about the Math For a babysitting course, Toma is designing a toy that will be filled with water. ? How many millilitres of water will Toma’s toy hold? 1 Understand the Problem To determine the number of millilitres of water the toy will hold, Toma needs to determine its volume in cubic centimetres. (1 mL 1 cm3) 3.0 cm 6.0 cm 2 Make a Plan Toma will use diagrams and a formula to determine the volume of each of the three figures that make up the toy. Then she will add the three volumes. 3.5 cm 3 Carry Out the Plan Figure Area of base of figure 3.0 cm top 6.0 cm 4.0 cm middle 5.0 cm bottom 3.5 cm 4.0 cm 4.0 cm 6.0 cm 4.0 cm 5.0 cm 6.0 cm Volume of figure Area of base of top figure r 2 (1.5 cm)2 7.1 cm2 Volume of top figure area of base height 7.1 cm2 6.0 cm 42.6 cm3 Area of base of middle figure r 2 (2.0 cm)2 12.6 cm2 Volume of middle figure area of base height 12.6 cm2 5.0 cm 63.0 cm3 The area of the base of the bottom figure is the area of six triangles. Area of one triangle 1 base height 2 1 (4.0 cm) 3.5 cm 2 7.0 cm2 Volume of bottom figure area of base height 42.0 cm2 6.0 cm 252.0 cm3 Area of six triangles 6 7.0 cm2 42.0 cm2 382 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:22 AM Page 383 Total volume of toy 42.6 cm3 63.0 cm3 252.0 cm3 357.6 cm3 The volume of Toma’s toy is about 358 cm3. Since 1 cm3 1 mL, the toy will hold about 358 mL of water. 4 Look Back Toma estimates that the top two cylinders are smaller than a 4 cm by 4 cm by 11 cm square prism that has a volume of 176 cm3. She thinks the bottom is smaller than a 6 cm cube that has a volume of 216 cm3. Her toy volume calculation is less than 176 cm3 216 cm3 392 cm3, so her answer is reasonable. Reflecting 1. How did the strategy of using diagrams help Toma solve this problem? 2. How did using diagrams to compare the sizes of the figures help Toma check her answer? Work with the Math Example: Using a tree diagram to determine probability Maria and Tran are playing a game with two standard dice. To win the game, they need to roll a sum of 8. What is the probability of rolling a sum of 8? Tran’s Solution 1 Understand the Problem I have to determine the number of different ways that I can roll a sum of 8 and compare this number with the number of possible outcomes. 2 Make a Plan I’ll draw a tree diagram to list all the possible outcomes. Then I’ll count the number of outcomes that give a sum of 8. 3 Carry Out the Plan Roll 1 1 2 3 4 Roll 2 Sum 123456 234567 123456 345678 123456 456789 12345 6 5 6 7 8 9 10 5 1234 5 6 6 7 8 9 10 11 6 123 4 5 6 7 8 9 10 11 12 There are 36 possible outcomes. The probability of rolling a sum of 8 is 5. 36 4 Look Back I see a pattern in the sums. In each group, they increase by 1. So, I’m sure that my diagram shows all the possible combinations. NEL Geometry and Measurement Relationships 383 *EMATH8_C11_v3b A 8/31/05 9:23 AM Page 384 6. Erik rolls two standard dice. Determine the probability that the sum will be 6, 7, or 8. Checking 3. Sketch the prisms that make up this cabin. Then calculate the volume and surface area of the whole cabin. 0.5 m 7. Fritz is making a stained-glass window. It consists of a rectangle that is 0.5 m wide by 2.5 m long, with a semicircle above the rectangle. a) Sketch an outline of the window. Label the dimensions. b) How much glass does Fritz need? 2.2 m 2.0 m 8. To copy a poster, Sohel reduces each dimension by 70%. The width of the original poster is 0.4 m. What is the width of the reduced copy? 9. How many squares are in figure 100? 1.8 m B Practising 4. Sketch the prisms that make up this skateboard ramp. Then calculate the volume and surface area of the whole skateboard ramp. Figure 1 1.6 m Figure 3 10. How many different ways can 360 players in a marching band be arranged in a rectangle? 1.6 m 4.2 m 2.1 m 1.2 m 5. Perdita has a red shirt, a black shirt, a yellow shirt, and a white shirt. She also has a pair of white shorts, a pair of red shorts, and a pair of blue shorts. a) Determine the number of different outfits that Perdita could wear. b) What is the probability that she will wear at least one piece of clothing that is red? 384 Chapter 11 Figure 2 11. A city park is a square with 600 m sides. Diane started walking from a point 150 m south of the northwest corner, straight to a point 150 m north of the southwest corner. How far did she walk? 12. James is estimating the amount of paint he needs for the walls of his 3.4 m by 2.6 m bedroom. His bedroom is 2.7 m high. One litre of paint covers about 10 m2. About how much paint does James need? NEL *EMATH8_C11_v3b 8/31/05 9:23 AM Page 385 CALCULATING SURFACE AREA OF CUBE STRUCTURES Use nine linking cubes to build this shape. You will need • linking cubes Each linking cube has a surface area of 6 square units. So, the total surface area of nine unattached cubes is 9 6 54 square units. In your shape, however, some of the cubes are attached to each other. Therefore, the surface area of your shape is less than 54 square units. 1. How many pairs of faces are attached to each other? 2. How can you use your answer to question 1 to calculate the surface area of your shape? 3. Build each shape below, and calculate its surface area. NEL a) c) b) d) Geometry and Measurement Relationships 385 *EMATH8_C11_v3b 8/31/05 9:23 AM Page 386 Mid-Chapter Review Frequently Asked Questions Q: How do you calculate the surface area of a cylinder? A: Sketch a net of the cylinder. The surface area is the sum of the areas of the faces. Like the base and top of a rectangular prism, the base and top of a cylinder are congruent. Therefore, they have the same area. Calculate the area of the base, and double it. Add this to the area of the curved surface. top 4.0 cm 4.0 cm 12.0 cm curved surface 12.0 cm d base Surface area 2 area of base area of curved surface 2 r 2 (d h) 2 (4.0 cm)2 ( 8.0 cm 12.0 cm) 402.1 cm2 Q: How do you calculate the volume of a cylinder? A: The base of a cylinder is a circle. Calculate the volume of a cylinder in the same way you would calculate the volume of a prism—multiply the area of the base by the height. Volume area of base height r 2 h (4.0 cm)2 12.0 cm 603.2 cm3 386 Chapter 11 4.0 cm 12.0 cm NEL *EMATH8_C11_v3b 8/31/05 9:23 AM Page 387 Practice Questions (11.2) 1. Sketch a net for each cylinder, and label its dimensions. Then calculate the surface area. a) b) c) d) (11.2) (11.2) (11.2) Item Radius of base (cm) Height of cylinder (cm) potato-chip container coffee can CD case oil barrel 4 7.5 8.5 25.0 8 15.0 20.5 80.0 2. Karim is painting a design on a cylindrical barrel. The height of the barrel is 1.2 m. The radius of its base is 0.3 m. What area will the paint have to cover? (Remember to include the bottom and lid of the barrel.) 3. Write step-by-step instructions for determining the surface area of an empty paper-towel roll. 4. Determine the surface area of each tin. a) 170 mm b) 12.0 cm (11.3) 8. Determine the volume of this figure. (11.3) Explain what you did. 2.5 m 8.0 m 6.5 m 4.0 m 12.0 m 9. A soup can has a radius of 4 cm and a height of 11 cm. There are 24 cans in one case. How many litres of soup are in one case? (11.3) 12.0 cm 60 mm (11.2) 7. Deirdre is buying birdseed for the class bird feeder. The bird feeder is a cylinder with a diameter of 25 cm and a height of 45 cm. How many millilitres of seed (11.3) should she buy? 5. A cylindrical candle has a radius of 6 cm and a height of 20 cm. How much waxed paper will Jake need to wrap the candle? 10. The height of each cylinder in a set of food-storage containers is 30 cm. The radius of the largest container is 10 cm. The volume of the smallest container is 1 3 the volume of the largest container. The volume of the middle-sized container is 2 3 the volume of the largest container. What (11.4) is the volume of each container? 10 cm 6. Determine the volume of a cylinder that is 20 cm high and has the radius or diameter below. a) radius 13 cm b) radius 6.5 cm c) diameter 20 cm NEL 30 cm Geometry and Measurement Relationships 387 *EMATH8_C11_v3b 8/31/05 9:23 AM Page 388 You will need 11.5 Exploring the Platonic Solids GOAL Investigate properties of the Platonic solids. • congruent equilateral triangles • congruent squares • congruent regular pentagons • congruent regular hexagons • tape • a protractor Explore the Math A Platonic solid is a polyhedron with faces that are all congruent regular polygons. The same number of faces meet at all the vertices. There are only five Platonic solids. tetrahedron cube octahedron dodecahedron polyhedron a 3-D shape that has polygons as its faces icosahedron ? Why are there only five Platonic solids? A. Look at the five Platonic solids. What shapes are the faces? How many faces meet at each vertex? 388 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:24 AM Page 389 B. A Platonic solid can have faces that are equilateral triangles. What is the measure of the interior angles of an equilateral triangle? C. Using only equilateral triangles, draw the nets for as many polyhedrons as you can. Make sure that the same number of triangles meet at each vertex. Cut out and fold your nets to make sure that they work. D. For each polyhedron you made, how many faces meet at each vertex? What is the sum of the angles at each vertex? E. What is the least number of equilateral triangles you can join at a vertex and still fold the net to make a Platonic solid? What is the greatest number? F. Repeat steps B to E using only squares. G. Repeat steps B to E using only regular pentagons. H. Copy and complete the following chart. Face information Platonic solid Polygon tetrahedron cube octahedron dodecahedron icosahedron equilateral triangle square equilateral triangle regular pentagon equilateral triangle Measure of interior angles on one face Number of faces at each vertex Sum of angles at each vertex Explain why you cannot use any other regular polygons as the faces of a Platonic solid. Explain why you cannot use any more of the regular polygons you have already used. I. Reflecting 1. Why does a Platonic solid look the same no matter which vertex you position at the top? 2. Refer to your chart. Look at the measures of the interior angles in each Platonic solid and the number of faces. How does the measure of the interior angles determine the number of faces that can join at a vertex? 3. A Platonic solid cannot be made from regular polygons that have more than five sides. How can you use the measures of the interior angles of regular polygons to show this is true? NEL Geometry and Measurement Relationships 389 *EMATH8_C11_v3b 8/31/05 9:24 AM Page 390 You will need 11.6 Polyhedron Faces, Edges, • pipe cleaners or straws • modelling clay and Vertices GOAL Determine how the number of faces, edges, and vertices of a polyhedron are related. Learn about the Math Toma and Benjamin noticed that whenever you make a 2-D polygon, the number of vertices is the same as the number of edges. They wondered whether the number of faces, vertices, and edges of 3-D polyhedrons are related. pattern links the number of faces, edges, and ? What vertices of a polyhedron? Benjamin tried building an unusual polyhedron first. “I’ll start building it from the top. The number of vertices and number of edges are the same, and there is one face.” Part built Number of faces Number of vertices top 1 4 390 Chapter 11 Number of edges 4 NEL *EMATH8_C11_v3b 8/31/05 9:24 AM Page 391 “Next, I’ll add squares. For each square, I add one new face, two new vertices, and three new edges. The total number of new faces and new vertices is equal to the number of new edges. “Now I’ll add triangles. For each triangle, I add one new face and one new edge, but no new vertices. Again, the total number of new faces and vertices is the same as the number of new edges.” Part built top 4 new squares 4 new triangles Number of faces Number of vertices Number of edges 1 4 more 4 more 4 8 more 0 more 4 12 more 4 more A. Construct the parts that Benjamin constructed. Add the next set of squares. Explain why the number of edges is 1 less than the total number of faces and vertices. B. Add the bottom of the polyhedron. Explain why you have added no new edges or vertices, but one new face. C. Why is the number of edges 2 less than the total number of faces and vertices? D. Choose one of the following shapes. Compare the number of edges with the total number of faces and vertices. What do you observe? E. Compare your results with the results of students who chose different shapes. What do you notice? Reflecting 1. The relationship you described in step C is called Euler’s formula (pronouced “oiler”). Explain why it can be written as F V E 2, where F is the number of faces, V is the number of vertices, and E is the number of edges of the shape. 2. How does Euler’s formula allow you to predict the number of edges, faces, or vertices of a shape if you know two of these values? NEL Geometry and Measurement Relationships 391 *EMATH8_C11_v3b 8/31/05 9:25 AM Page 392 Work with the Math Example 1: Checking whether a polyhedron is possible Is it possible to make a polyhedron with 6 faces, 7 vertices, and 10 edges? Tran’s Solution FVE2 F V E 6 7 10 3 I used Euler’s formula. If it is possible to make a polyhedron like this, the result should be 2 when I substitute the values into Euler’s formula. I substituted the values into the formula. The result is 3, not 2, so it is not possible to make such a polyhedron. Example 2: Using Euler’s formula to determine a missing value If a polyhedron has 10 faces and 18 edges, how many vertices should it have? Benjamin’s Solution FVE2 10 V 18 2 V82 V8828 V 10 A I used Euler’s formula. I substituted 10 for the number of faces and 18 for the number of edges. I used balancing to solve the equation. The polyhedron should have 10 vertices. Checking 3. A student used 10 pipe cleaners to make the edges of a polyhedron. If the polyhedron has 6 vertices, how many faces must it have? 4. Show that Euler’s formula works for a tetrahedron. 392 Chapter 11 B Practising 5. Show that Euler’s formula works for the other four Platonic solids: a cube, an octahedron, a dodecahedron, and an icosahedron. NEL *EMATH8_C11_v3b 8/31/05 9:25 AM Page 393 6. Copy and complete the chart for some polyhedrons. Number of faces Number of edges Number of vertices 9 12 5 6 6 20 16 7 10. Make another cube using modelling clay. Then make a pyramid on each face of the cube. Show that Euler’s formula works for this polyhedron. 11. Imagine that you drilled a rectangular hole through a cube. Does Euler’s formula work for the new shape? 30 12 10 6 12. a) Construct a triangular prism. b) c) d) e) 7. The following crystals and gemstones have been cut to form polyhedrons. Show that Euler’s formula works for each polyhedron. a) How many faces does the prism have? How many edges does the prism have? How many vertices does the prism have? Show that Euler’s formula works for the prism. 13. Repeat question 12 using a pentagonal pyramid. C b) Extending 14. Make a cube using modelling clay. Mark a point in the centre of each face. Imagine that you joined these points with string inside the cube to form a polyhedron. Show that Euler’s formula works for this polyhedron. 15. A prism has a base with n sides. 8. Show that Euler’s formula works for this cuboctahedron. a) b) c) d) How many faces does the prism have? How many edges does the prism have? How many vertices does the prism have? Show that Euler’s formula works for the prism. 16. A pyramid has a base with n sides. 9. Make a cube using modelling clay. Cut the corners off the cube. Show that Euler’s formula works for the new shape. NEL a) How many faces does the pyramid have? b) How many edges does the pyramid have? c) How many vertices does the pyramid have? d) Show that Euler’s formula works for the pyramid. Geometry and Measurement Relationships 393 *EMATH8_C11_v3b 8/31/05 9:25 AM Page 394 THE VOLUMIZER GAME! In this game, you will calculate the volume of a cylinder using a radius given on a card and a height obtained by rolling a die or spinning a spinner. You will need • index cards • a die or spinner • a calculator Number of players: 2 or more Rules 1. To create a deck of Volumizer cards, use about 20 blank index cards. On each card, sketch a cylinder and 6 label the radius of the base in centimetres. 2. To play the game, each player 5 Volumizer 1 2 • selects a Volumizer card from the pile 4 3 • rolls the die or spins the spinner • calculates the volume of the cylinder, using the number on the die or spinner as the height of the cylinder in centimetres All players take their turns at the same time. Players may check each other’s calculations. r 12.0 cm 3. The player who has the cylinder with the greatest volume keeps the card. The other players return their cards to the deck. 4. The game is finished when the number of cards left in the pile is less than the number of players. 5. The player with the most cards at the end of the game is the winner. 394 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:26 AM Page 395 Chapter Self-Test 1. Draw a net for the paper needed to wrap each candle. Label the dimensions. 4. This railway car is 320 cm in diameter and 17.2 m long. Calculate its volume. 4 cm 4 cm 14 cm 11 cm 2. Which parts of this net of a cylinder are equal to the circumference of the base? c a b b a d 5. Suppose that you increase the height of this cylinder by 10 cm. By how much does the volume increase? 5 cm 3. Calculate the surface area and volume of each cylinder. a) 5.1 cm c) 6 cm 7 cm 2.6 cm 9 cm b) 4.1 cm d) 5.0 cm 7.0 cm 16.0 cm NEL 6. No more than three congruent squares can meet at each vertex of a Platonic solid. Explain why. 7. A soccer ball is a polyhedron made from 20 regular hexagons and 12 regular pentagons. It has 60 vertices. Determine the number of edges on a soccer ball. Geometry and Measurement Relationships 395 *EMATH8_C11_v3b 8/31/05 9:26 AM Page 396 Chapter Review Frequently Asked Questions Q: What is a Platonic solid? A: A Platonic solid is a polyhedron with faces that are all congruent regular polygons. The same number of faces meet at all the vertices in a Platonic solid. The five Platonic solids are shown below. tetrahedron cube octahedron dodecahedron icosahedron Q: Why are there only five Platonic solids? A: The total of the interior angles that meet at each vertex of a Platonic solid must be less than 360°. At least three faces must meet at each vertex. Regular polygons with more than five sides have angles that measure at least 120°. When you build a polyhedron, at least three faces have to meet at each vertex to make the polyhedron 3-D. If you tried to use the faces of polygons with more than five sides as the faces of a polyhedron, the sum of the angles that meet at each vertex would be at least 360°. This is not possible. Q: How are the number of edges, vertices, and faces of a polyhedron related? A: The relationship among the number of edges, vertices, and faces can be represented using the equation F V E 2, where F is the number of faces, E is the number of edges, and V is the number of vertices. This equation is known as Euler’s formula. For example, the following cube has 6 faces, 8 vertices, and 12 edges. FVE2 6 8 12 2 396 Chapter 11 NEL *EMATH8_C11_v3b 8/31/05 9:26 AM Page 397 Practice Questions (11.2) 1. Calculate the surface area of this cylinder. 3.3 m 6.2 m (11.3) 2. Calculate the volume of the cylinder in question 1. 3. Mohammed is choosing a bass drum to buy for his band. The “Bashmaster” is 71.1 cm in diameter and 35.6 cm high. The “Crash” is 91.5 cm in diameter and 66.0 cm high. The “Boomalot” is 81.3 cm in diameter and 45.7 cm high. 4. A glass in the shape of a cylinder is 10.0 cm high and has a diameter of 3.5 cm. How many millilitres of juice will the glass hold (11.3) if it is filled to the top? 5. What might be the dimensions of a cylindrical container that holds 750 mL (11.3) of juice? 6. Sketch a shape made up of a cylinder and a triangular prism that has a total volume (11.4) between 100 cm3 and 200 cm3. 7. Why is it impossible to have a Platonic solid in which six or more equilateral (11.5) triangles meet at each vertex? 8. Show that Euler’s formula works for a (11.6) pentagonal prism. a) Which drum has the greatest surface area? Justify your answer. b) Which drum has the greatest volume in cubic centimetres? Justify your answer. (11.2) (11.3) 9. A polyhedron has 9 edges and 6 vertices. a) Calculate the number of faces. b) Sketch the polyhedron. (11.6) 10. A polyhedron has 6 faces and 6 vertices. (11.6) Calculate the number of edges. 11. A polyhedron has 8 faces and 12 edges. (11.6) Calculate the number of vertices. NEL Geometry and Measurement Relationships 397 *EMATH8_C11_v3b 8/31/05 9:27 AM Page 398 Chapter Task Storage Capacity of a Silo In this task, you will design a silo that can be used to store corn for animal feed. The outside of the silo will be painted to make it rust resistant and more attractive. Keep in mind: • The paint comes in 3.8 L cans. Each can covers an area of 40 m2 and costs $35, including taxes. • In 2003, corn for animal feed was sold for about $120 per tonne. can you design a silo and ? How report on the costs? A. Sketch the silo you recommend. Show its diameter and height. B. Calculate the surface area of your silo. C. Calculate the cost to cover your silo with one coat of paint. D. What is the volume of the corn that can be stored in your silo? E. What mass of corn can be stored in your silo? Use the height of the corn in your silo and the following table to estimate the mass of the corn. (1 t 1000 kg) Height of corn (m) Mass (kg) of 1 m3 of corn 9 12 15 18 21 24 570 610 660 700 740 770 F. Estimate the value of the corn that can be stored in your silo. G. Prepare a written report that shows your calculations and explains your thinking. 398 Chapter 11 Task Checklist ✓ ❏ Did you show all your steps? ✓ ❏ Did you explain your thinking? ✓ ❏ ✓ ❏ Did you draw and label your diagram neatly and accurately? Did you use appropriate math vocabulary? NEL

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