11-1 Space Figures and Cross Sections 11-1 1. Plan Objectives 1 2 To recognize polyhedra and their parts To visualize cross sections of space figures Examples 1 2 3 4 5 Identifying Vertices, Edges, and Faces Using Euler’s Formula Verifying Euler’s Formula Describing a Cross Section Drawing a Cross Section GO for Help What You’ll Learn Check Skills You’ll Need • To recognize polyhedra and For each exercise, make a copy of the cube at the right. Shade the plane that contains the indicated points. 1–7. See back of book. their parts • To visualize cross sections of space figures . . . And Why To learn about medical techniques, as in Exercise 44. 1. A, B, and C 2. A, B, and G 3. A, C, and G 4. A, D, and G 5. F, D, and G 6. B, D, and G Lesson 1-3 B C D A F 7. the midpoints of AD, CD, EH, and GH G E H New Vocabulary • polyhedron • face • edge • vertex • cross section Math Background The references to plane figures are somewhat informal. Explain, for example, that a base of a cylinder is not technically a circle; it is a circle together with the circle’s interior. Similarly, a face of a polyhedron is not actually a polygon; it is a polygon together with its interior. 1 Identifying Parts of a Polyhedron A polyhedron is a three-dimensional figure whose surfaces are polygons. Each polygon is a face of the polyhedron. An edge is a segment that is formed by the intersection of two faces. A vertex is a point where three or more edges intersect. Vocabulary Tip Polyhedron comes from the Greek poly for “many” and hedron for “side.” A cube is a polyhedron with six sides, or faces, each of which is a square. More Math Background: p. 596C 1 EXAMPLE Faces Edge Vertex Identifying Vertices, Edges, and Faces a. How many vertices are there in the polyhedron at the right? List them. Lesson Planning and Resources H There are five vertices: D, E, F, G, and H. See p. 596E for a list of the resources that support this lesson. There are eight edges: DE , EF, FG, GD, DH, EH, FH, and GH. PowerPoint F G b. How many edges are there? List them. D E c. How many faces are there? List them. Bell Ringer Practice There are five faces: #DEH, #EFH, #FGH, #GDH, and the quadrilateral DEFG. R Check Skills You’ll Need For intervention, direct students to: Quick Check Identifying Planes Lesson 1-3: Example 4 Extra Skills, Word Problems, Proof Practice, Ch. 1 1 List the vertices, edges, and faces of the polyhedron. R, S, T, U, V; RS, RU, RT , VS, VU, VT , SU, UT , TS; kRSU, kRUT, kRTS, kVSU, kVUT, kVTS S T U V 598 Chapter 11 Surface Area and Volume Special Needs Below Level L1 Review nets by having students cut-out various nets and form their corresponding three-dimensional figures. Clarify that there are many possible nets for the same polyhedron. 598 learning style: tactile L2 Some students may think that spheres and cylinders are polyhedrons. Emphasize that the surfaces of polyhedrons are polygons, whose sides must be line segments. Have students draw examples of polygons on the board. learning style: visual 2. Teach Leonhard Euler, a Swiss mathematician, discovered a relationship among the numbers of faces, vertices, and edges of any polyhedron. The result is known as Euler’s Formula. Key Concepts Formula Guided Instruction Euler’s Formula The numbers of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F + V = E + 2. 2 EXAMPLE 1 Using Euler’s Formula PowerPoint Additional Examples 1 How many vertices, edges, and faces of the polyhedron are there? List them. The polyhedron has 2 hexagons and 6 rectangles for a total of 8 faces. The 2 hexagons have a total of 12 edges. The 6 rectangles have a total of 24 edges. If the hexagons and rectangles are joined to form a polyhedron, each edge is shared by two faces. Therefore, the number of edges in the polyhedron is one half of the total of 36, or 18. Connection Euler’s Formula 8 + V = 18 + 2 Substitute. V = 12 Euler’s Formula applies to the polyhedron suggested by the panels on a volleyball. Quick Check F+V=E+2 A E J D I C G H 10 vertices, 15 edges, and 7 faces; A, B, C, D, E, F, G, H, I, J; AF, BG, CH, DI, EJ, AB, BC, CD, DE, EA, FG, GH, HI, IJ, JF; ; pentagons ABCD and FGHIJ, and quadrilaterals ABGF, BCHG, CDIH, and EAFJ 2 Use Euler’s Formula to find the number of edges on a polyhedron with eight triangular faces. 12 edges In two dimensions, Euler’s Formula reduces to F+V=E+1 where F is the number of regions formed by V vertices linked by E segments. EXAMPLE B F Simplify. Count the number of vertices in the figure to verify the result. 3 Teaching Tip Encourage students to work systematically as they list the vertices, edges, and faces. Count faces and edges. Then use Euler’s Formula to find the number of vertices in the polyhedron at the right. Real-World EXAMPLE 2 Use Euler’s Formula to find the number of edges of a polyhedron with 6 faces and 8 vertices. 12 edges Verifying Euler’s Formula Verify Euler’s Formula for a two-dimensional net of the solid in Example 2. 3 Using the pentagonal prism in Additional Example 1, verify Euler’s Formula. Then draw a net for the figure and verify Euler’s Formula for the two-dimensional figure. 7 ± 10 ≠15 ± 2 Draw a net: Count the regions: F = 8 Count the vertices: V = 22 Count the segments: E = 29 8 + 22 = 29 + 1 Quick Check 3 The figure at the right is a trapezoidal prism. a. Verify Euler’s formula F + V = E + 2 for the prism. 6 + 8 = 12 + 2 b. Draw a net for the prism. See margin. c. Verify Euler’s formula F + V = E + 1 for your two-dimensional net. Sample: 6 + 14 = 19 + 1 7 ± 18 ≠24 ± 1 Quick Check Lesson 11-1 Space Figures and Cross Sections Advanced Learners 599 3. English Language Learners ELL L4 Have students determine if values of F, V, and E that satisfy Euler’s Formula make an existing polyhedron. learning style: verbal Make sure students understand the difference between polyhedron and polygon. Show models of different polyhedrons and cutouts of different polygons. learning style: visual 599 Guided Instruction 2 1 Describing Cross Sections Error Prevention! Students may think the plane of a cross section must be horizontal or vertical. Show a cross section of an apple or orange cut along a plane that is neither horizontal nor vertical. 5 EXAMPLE A cross section is the intersection of a solid and a plane. You can think of a cross section as a very thin slice of the solid. Sclera Lens Cornea Retina Iris Optic nerve Blood vessels Pupil 4 Teaching Tip Point out that the example assumes that the bottom face of the cube is horizontal. Ask: If the cube were tilted slightly, what might the cross section look like? Sample: parallelogram Real-World Connection EXAMPLE Describing a Cross Section Describe each cross section. a. Cross sections are used to study the anatomy of the eye. b. Visual Learners The cross section is a square. Encourage students to slice cubes of butter, ice cream, or modeling clay at home to investigate how planes may intersect cubes. Quick Check 4. Size of sketches may vary, Samples: PowerPoint Additional Examples The cross section is a triangle. 4 For the funnel shown, sketch each of the following. a. a horizontal cross section a–b. See left. b. a vertical cross section that contains the axis of symmetry a. To draw a cross section, you can sometimes use the idea from Postulate 1-3 that the intersection of two planes is exactly one line. 4 Describe this cross section. b. 5 EXAMPLE Drawing a Cross Section Visualization Draw and describe a cross section formed by a vertical plane intersecting the front and right faces of the cube. triangle 5 Draw and describe a cross section formed by a vertical plane intersecting the top and bottom faces of a cube. Check students’ work; square or rectangle. Resources • Daily Notetaking Guide 11-1 A vertical plane cuts the vertical faces of the cube in parallel segments. Draw the parallel segments. 5. square Join their endpoints. Shade the cross section. L3 • Daily Notetaking Guide 11-1— L1 Adapted Instruction Closure What is a polyhedron and how is Euler’s Formula related to it? A polyhedron is a threedimensional figure whose surfaces are polygons. Euler’s Formula relates the number of faces (F), vertices (V), and edges (E) of a polyhedron such that F ± V ≠E ± 2. 600 The cross section is a rectangle. Quick Check 600 5 Draw and describe the cross section formed by a horizontal plane intersecting the left and right faces of the cube. See left. Chapter 11 Surface Area and Volume 17. rectangle 18. square 19. rectangle EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. 3. Practice Practice and Problem Solving Assignment Guide Practice by Example Example 1 For each polyhedron, how many vertices, edges, and faces are there? List them. M 1. 2. 4, 6, 4 (page 598) P (page 599) U P D V F E O 3. C A 1–3. See back of book for lists. N Example 2 B Q G H S 8, 12, 6 R X W Y (page 599) Example 4 (page 600) 11. 56-60 61-68 To check students’ understanding of key skills and concepts, go over Exercises 8, 16, 22, 30, 32. 6. Faces: 20 12 Edges: 30 Vertices: j Exercises 13–15 As a class, explore how the shape of the cross section changes with the orientation of the plane. Verify Euler’s Formula for each polyhedron. Then draw a net for the figure and verify Euler’s Formula for the two-dimensional figure. 10–12. See back of book. 10. 13-19, 21-26, 39-45 C Challenge 46-55 Homework Quick Check 5. Faces: 8 12 Edges: j Vertices: 6 Use Euler’s Formula to find the number of vertices in each polyhedron described below. 5 9 7. 6 square faces 8. 5 faces: 1 rectangle 9. 9 faces: 1 octagon 8 and 4 triangles and 8 triangles Example 3 2 A B Test Prep Mixed Review T 10, 15, 7 Use Euler’s Formula to find the missing number. 4. Faces: j 8 Edges: 15 Vertices: 9 1 A B 1-12, 20, 27-38 12. Exercise 19 Draw a cube on the board, using a different color for each set of opposite edges. Point out that opposite edges are parallel and are not on the same face. Describe each cross section. 13. two concentric circles 13. 14. 15. rectangle GPS Guided Problem Solving L3 L4 Enrichment triangle 16. For the nut shown, sketch each of following. a. a horizontal cross section a–b. See back of book. b. a vertical cross section that contains the vertical line of symmetry L2 Reteaching L1 Adapted Practice Practice Name Class L3 Date Practice 10-1 Space Figures and Nets 1. Choose the nets that will fold to make a cube. A. Example 5 (page 600) Visualization Draw and describe a cross section formed by a vertical plane intersecting the cube as follows. 17–19. See margin. 17. The vertical plane intersects the front and left faces of the cube. B. C. D. Draw a net for each figure. Label each net with its appropriate dimensions. 2. 16 cm 7 cm 2 cm 3. 8 cm 4. 1 cm 2 cm 32 cm 1 cm 40 cm Match each three-dimensional figure with its net. 5. 6. 7. 8. 18. The vertical plane intersects opposite faces of the cube. A. 19. The vertical plane contains opposite edges of the cube. B. C. D. 9. Choose the nets that will fold to make a pyramid with a square base. A. Lesson 11-1 Space Figures and Cross Sections B. C. D. 601 © Pearson Education, Inc. All rights reserved. A Use Euler’s Formula to find the missing number. 10. Faces: 5 Edges: 7 Vertices: 5 11. Faces: 7 Edges: 9 Vertices: 6 12. Faces: 8 Edges: 18 Vertices: 7 601 Exercise 38 Ask: What do you know about a cube that might help you solve this problem? A cube has 6 square faces. Connection to Calculus B Apply Your Skills 21. rectangle Exercises 27–29 Formulas for the volumes of more complicated solids of revolution are developed in calculus. 20. a. Open-Ended Sketch a polyhedron whose faces are all rectangles. Label the lengths of its edges. a–b. See back of book. b. Use graph paper to draw two different nets for the polyhedron. Visualization Draw and describe a cross section formed by a plane intersecting the cube as follows. 21–23. See left. 21. The plane is tilted and intersects the left and right faces of the cube. 22. The plane contains opposite horizontal edges of the cube. Connection to Astronomy 22. rectangle 23. The plane cuts off a corner of the cube. Exercise 30 Early scientists used Describe the cross section shown. triangle circle 24. 25. Platonic solids to attempt to explain the universe. Have students investigate some of these explanations. 26. 2 trapezoids 23. triangle Visualization A plane region that revolves completely about a line sweeps out a solid of revolution. Use the sample to help you describe the solid of revolution you get by revolving each region about line <. Sample: Revolve the rectangular region about the line / and you get a cylinder as a solid of revolution. 27. ᐉ ᐉ 28. ᐉ ᐉ cylinder attached to a cone 29. ᐉ sphere cone Sports Equipment Some balls are made from panels that suggest polygons. The ball then suggests a polyhedron to which Euler’s Formula, F ± V ≠E ± 2, applies. 30. A soccer ball suggests a polyhedron with 20 regular hexagons and 12 regular pentagons. How many vertices does this polyhedron have? 60 31. Show how Euler’s Formula applies to the polyhedron suggested by the volleyball pictured on page 599. (Hint: It has 6 sets of 3 panels.) 18 + 32 = 48 + 2 Euler’s Formula F ± V ≠E ± 1 applies to any two-dimensional network where F is the number of regions formed by V vertices linked by E edges (or paths). Verify Euler’s Formula for each network shown. 46. GPS 32. 47. GO 48. 49. 602 50. 34. nline Homework Help Visit: PHSchool.com Web Code: aue-1101 602 33. 6+4=9+1 4+6=9+1 35. Draw a network of your own. Verify Euler’s Formula for it. Check students’ work. Chapter 11 Surface Area and Volume 51. 52. 53. 54. 5+5=9+1 36. There are five regular polyhedrons. They are called regular because all their faces are congruent regular polygons, and the same number of faces meet at each vertex. They are also called Platonic Solids after the Greek philosopher Plato (427–347 B.C.). 4. Assess & Reteach PowerPoint Lesson Quiz 1. Draw a net for the figure. Tetrahedron Octahedron Hexahedron Real-World Connection A fluorite crystal forms as a regular octahedron. Icosahedron Sample: Dodecahedron a. Match each net below with a Platonic Solid. A. B. C. D. E. A. icosahedron B. octahedron C. tetrahedron D. hexahedron 36b. regular triangular pyramid, cube E. dodecahedron b. The first two Platonic solids have more familiar names. What are they? c. Verify that Euler’s Formula is true for the first three Platonic solids. 4 + 4 = 6 + 2, 6 + 8 = 12 + 2, 8 + 6 = 12 + 2 37. Multiple Choice A cube has a net with area 216 in.2. How long is an edge of the cube? A 6 in. 15 in. 36 in. 54 in. Use Euler’s Formula to solve. 2. A polyhedron with 12 vertices and 30 edges has how many faces? 20 3. A polyhedron with 2 octagonal faces and 8 rectangular faces has how many vertices? 16 4. Describe the cross section. Draw each object. Then draw a horizontal and a vertical cross section. 38. a golf tee 39. a football 40. a baseball bat 41. a banana 42. a pear 43. a bagel 38–43. Check students’ work. 44. Writing Cross sections are used in medical training and research. Research and write a paragraph on how magnetic resonance imaging (MRI) is used to study cross sections of the brain. Check students’ work. C Challenge 45. Draw a solid that has the following cross sections. 45. circle 5. Draw and describe a cross section formed by a vertical plane cutting the left and back faces of a cube. Check students’ drawings; rectangle. Alternative Assessment horizontal vertical Visualization Draw a plane intersecting a cube to get the cross section indicated. 46. scalene triangle 47. isosceles triangle 48. equilateral triangle 49. trapezoid 50. isosceles trapezoid 51. parallelogram 52. rhombus lesson quiz, PHSchool.com, Web Code: aua-1101 53. pentagon 46–54. See margin. Have each student bring a realworld polyhedron to class. Have them verify Euler’s Formula and then draw a net for the solid. 54. hexagon Lesson 11-1 Space Figures and Cross Sections 603 603 Test Prep Test Prep Resources Multiple Choice For additional practice with a variety of test item formats: • Standardized Test Prep, p. 657 • Test-Taking Strategies, p. 652 • Test-Taking Strategies with Transparencies For Exercises 55–56, you may need Euler’s Formula, F + V = E + 2. 55. A polyhedron has four vertices and six edges. How many faces does it have? B A. 2 B. 4 C. 5 D. 10 56. A polyhedron has three rectangular faces and two triangular faces. How many vertices does it have? G F. 5 G. 6 H. 10 J. 12 57. The plane is horizontal. What best describes the shape of the cross section? D A. rhombus B. trapezoid C. parallelogram D. square 58. The plane is vertical. What best describes the shape of the cross section? J F. pentagon G. square H. rectangle J. triangle 59. [2] a. square Short Response b. Answers may vary. Sample: trapezoid 59. Draw and describe a cross section formed by a plane intersecting a cube as follows. a. The plane is parallel to a horizontal face of the cube. b. The plane cuts off two corners of the cube. a–b. See margin. Mixed Review [1] only 1 correct drawing GO for Help Lesson 10-8 60. Probability A shuttle bus to an airport terminal leaves every 20 min from a remote parking lot. Draw a geometric model and find the probability that a traveler who arrives at a random time will have to wait at least 8 min for the 60% bus to leave the parking lot. 0 4 8 12 16 20 61. Games A dartboard is a circle with a 12-in. radius. You throw a dart that hits the dartboard. What is the probability that the dart lands within 6 in. of the center of the dartboard? 25% Lesson 10-3 Lesson 8-3 Find the area of each equilateral triangle with the given measure. Leave answers in simplest radical form. 62. side 2 ft 63. apothem 8 cm 192"3 cm2 "3 ft2 Find the value of x to the nearest tenth. 65. 4.7 64. radius 100 in. 7500"3 in.2 66. 8.3 65Њ x x 6 36Њ 10 67. The lengths of the diagonals of a rhombus are 4 cm and 6 cm. Find the measures of the angles of the rhombus to the nearest degree. 67 and 113 604 604 Chapter 11 Surface Area and Volume
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