New Vocabulary polyhedron face edge vertex cross section EXAMPLE. Quick Check

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1 -. Plan Objectives To recognize polyhedra and their parts 2 To visualize cross sections of space figures xamples Identifying Vertices, dges, and aces 2 Using uler s ormula 3 Verifying uler s ormula 4 escribing a ross Section rawing a ross Section - What You ll Learn To recognize polyhedra and their parts To visualize cross sections of space figures... And Why To learn about medical techniques, as in xercise 44. Space igures and ross Sections heck Skills You ll Need O for elp or each exercise, make a copy of the cube at the right. Shade the plane that contains the indicated points. 7. See back of book.. A, B, and 2. A, B, and 3. A,, and 4. A,, and.,, and 6. B,, and 7. the midpoints of A,,, and New Vocabulary polyhedron face edge vertex cross section A B Lesson -3 Math Background The references to plane figures are somewhat informal. xplain, 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. More Math Background: p. 96 Identifying Parts of a Polyhedron Vocabulary Tip Polyhedron comes from the reek poly for many and hedron for side. A cube is a polyhedron with six sides, or faces, each of which is a square. A polyhedron is a three-dimensional figure whose surfaces are polygons. ach 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. Identifying Vertices, dges, and aces aces dge Vertex Lesson Planning and Resources See p. 96 for a list of the resources that support this lesson. a. ow many vertices are there in the polyhedron at the right? List them. There are five vertices:,,,, and. b. ow many edges are there? List them. There are eight edges:,,,,,,, and. Bell Ringer Practice heck Skills You ll Need or intervention, direct students to: Identifying Planes Lesson -3: xample 4 xtra Skills, Word Problems, Proof Practice, h. c. ow many faces are there? List them. There are five faces: #, #, #, #, and the quadrilateral. R List the vertices, edges, and faces of the polyhedron. R, S, T, U, V; RS, RU, RT, VS, VU, VT, SU, S UT, TS; krsu, krut, krts, kvsu, kvut, kvts U T 98 hapter Surface Area and Volume V 98 Special Needs L Review nets by having students cut-out various nets and form their corresponding three-dimensional figures. larify that there are many possible nets for the same polyhedron. learning style: tactile Below Level L2 Some students may think that spheres and cylinders are polyhedrons. mphasize that the surfaces of polyhedrons are polygons, whose sides must be line segments. ave students draw examples of polygons on the board. learning style: visual

2 Real-World Leonhard uler, a Swiss mathematician, discovered a relationship among the numbers of faces, vertices, and edges of any polyhedron. The result is known as uler s ormula. Key oncepts ormula uler s ormula onnection uler s ormula applies to the polyhedron suggested by the panels on a volleyball. The numbers of faces (), vertices (V), and edges () of a polyhedron are related by the formula + V = + 2. Using uler s ormula ount faces and edges. Then use uler s ormula to find the number of vertices in the polyhedron at the right. The polyhedron has 2 hexagons and 6 rectangles for a total of 8 faces. The 2 hexagons have a total of 2 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 8. + V = + 2 uler s ormula 8 + V = Substitute. V = 2 Simplify. ount the number of vertices in the figure to verify the result. Use uler s ormula to find the number of edges on a polyhedron with eight triangular faces. 2 edges In two dimensions, uler s ormula reduces to + V = + where is the number of regions formed by V vertices linked by segments. Verifying uler s ormula Verify uler s ormula for a two-dimensional net of the solid in xample 2. raw a net: ount the regions: = 8 ount the vertices: V = 22 ount the segments: = = 29 + The figure at the right is a trapezoidal prism. a. Verify uler s formula + V = + 2 for the prism = b. raw a net for the prism. See margin. c. Verify uler s formula + V = + for your two-dimensional net. Sample: = 9 + Lesson - Space igures and ross Sections Teach uided Instruction Teaching Tip ncourage students to work systematically as they list the vertices, edges, and faces. Additional xamples ow many vertices, edges, and faces of the polyhedron are there? List them. J A I 0 vertices, edges, and 7 faces; A, B,,,,,,, I, J; A, B,, I, J, AB, B,,, A,,, I, IJ, J; ; pentagons AB and IJ, and quadrilaterals AB, B, I, and AJ 2 Use uler s ormula to find the number of edges of a polyhedron with 6 faces and 8 vertices. 2 edges 3 Using the pentagonal prism in Additional xample, verify uler s ormula. Then draw a net for the figure and verify uler s ormula for the two-dimensional figure. 7 ± 0 ± 2 7 ± 8 24 ± 3. B Advanced Learners L4 ave students determine if values of, V, and that satisfy uler s ormula make an existing polyhedron. learning style: verbal nglish Language Learners LL Make sure students understand the difference between polyhedron and polygon. Show models of different polyhedrons and cutouts of different polygons. learning style: visual 99

3 uided Instruction rror 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. 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 ornea 2 escribing ross Sections Iris Pupil Real-World Lens Retina Blood vessels Sclera Optic nerve onnection ross sections are used to study the anatomy of the eye. 4 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. escribing a ross Section escribe each cross section. a. b. Visual Learners ncourage students to slice cubes of butter, ice cream, or modeling clay at home to investigate how planes may intersect cubes. 4 Additional xamples escribe this cross section. 4. Size of sketches may vary, Samples: a. 4 The cross section is a square. or 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 The cross section is a triangle. To draw a cross section, you can sometimes use the idea from Postulate -3 that the intersection of two planes is exactly one line. b. rawing a ross Section triangle raw and describe a cross section formed by a vertical plane intersecting the top and bottom faces of a cube. heck students work; square or rectangle. Resources aily Notetaking uide - L3 aily Notetaking uide - Adapted Instruction L. square Visualization raw and describe a cross section formed by a vertical plane intersecting the front and right faces of the cube. A vertical plane cuts the vertical faces of the cube in parallel segments. raw the parallel segments. Join their endpoints. Shade the cross section. losure What is a polyhedron and how is uler s ormula related to it? A polyhedron is a threedimensional figure whose surfaces are polygons. uler s ormula relates the number of faces (), vertices (V), and edges () of a polyhedron such that ± V ± hapter Surface Area and Volume The cross section is a rectangle. raw and describe the cross section formed by a horizontal plane intersecting the left and right faces of the cube. See left. 7. rectangle 8. square 9. rectangle 600

4 XRISS or more exercises, see xtra Skill, Word Problem, and Proof Practice. Practice and Problem Solving A Practice by xample xample (page 98) xample 2 (page 99) xample 3 (page 99) or each polyhedron, how many vertices, edges, and faces are there? List them.. M 2. B 3. U 4, 6, 4 A P V W 3. See back of R P book for lists. Q Y X N O S 8, 2, 6 T 0,, 7 Use uler s ormula to find the missing number. 4. aces: j 8. aces: aces: 20 2 dges: dges: j dges: 30 Vertices: 9 Vertices: 6 Vertices: j Use uler s ormula to find the number of vertices in each polyhedron described below square faces 8. faces: rectangle faces: octagon 8 and 4 triangles and 8 triangles Verify uler s ormula for each polyhedron. Then draw a net for the figure and verify uler s ormula for the two-dimensional figure See back of book Practice Assignment uide 2 A B -2, 20, A B 3-9, 2-26, 39-4 hallenge 46- Test Prep 6-60 Mixed Review 6-68 omework To check students understanding of key skills and concepts, go over xercises 8, 6, 22, 30, 32. xercises 3 As a class, explore how the shape of the cross section changes with the orientation of the plane. xercise 9 raw 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. xample 4 (page 600) escribe each cross section. 3. two concentric circles triangle 6. or 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 rectangle PS uided Problem Solving L3 nrichment Reteaching Adapted Practice L4 L2 L Practice Name lass ate Practice 0-. hoose the nets that will fold to make a cube. A. B... Space igures and Nets L3 xample (page 600) Visualization raw and describe a cross section formed by a vertical plane intersecting the cube as follows See margin. 7. The vertical plane intersects the front and left faces of the cube. 8. The vertical plane intersects opposite faces of the cube. 9. The vertical plane contains opposite edges of the cube. Lesson - Space igures and ross Sections 60 raw a net for each figure. Label each net with its appropriate dimensions cm 7 cm 8 cm 2 cm 2 cm 6 cm 32 cm cm 40 cm Match each three-dimensional figure with its net A. B hoose the nets that will fold to make a pyramid with a square base. A. B... Pearson ducation, Inc. All rights reserved. Use uler s ormula to find the missing number. 0. aces:. aces: 7 2. aces: 8 dges: 7 dges: 9 dges: 8 Vertices: Vertices: 6 Vertices: 7 60

5 xercise 38 Ask: What do you know about a cube that might help you solve this problem? A cube has 6 square faces. B Apply Your Skills 20. a. Open-nded 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. onnection to alculus xercises ormulas for the volumes of more complicated solids of revolution are developed in calculus. onnection to Astronomy xercise 30 arly scientists used Platonic solids to attempt to explain the universe. ave students investigate some of these explanations. 2. rectangle 22. rectangle 23. triangle Visualization raw and describe a cross section formed by a plane intersecting the cube as follows See left. 2. The plane is tilted and intersects the left and right faces of the cube. 22. The plane contains opposite horizontal edges of the cube. 23. The plane cuts off a corner of the cube. escribe the cross section shown. triangle circle 2 trapezoids 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. cylinder attached to a cone cone sphere Sports quipment Some balls are made from panels that suggest polygons. The ball then suggests a polyhedron to which uler s ormula, ± V ± 2, applies. 30. A soccer ball suggests a polyhedron with 20 regular hexagons and 2 regular pentagons. ow many vertices does this polyhedron have? Show how uler s ormula applies to the polyhedron suggested by the volleyball pictured on page 99. (int: It has 6 sets of 3 panels.) = uler s ormula ± V ± applies to any two-dimensional network where is the number of regions formed by V vertices linked by edges (or paths). Verify uler s ormula for each network shown. PS O nline omework elp Visit: PSchool.com Web ode: aue hapter Surface Area and Volume = = raw a network of your own. Verify uler s ormula for it. heck students work. + =

6 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 reek philosopher Plato ( B..). 4. Assess & Reteach Lesson Quiz. raw a net for the figure. Tetrahedron Octahedron Icosahedron exahedron odecahedron Sample: Real-World onnection A fluorite crystal forms as a regular octahedron. a. Match each net below with a Platonic Solid. A. B.... A. icosahedron B. octahedron. tetrahedron. hexahedron 36b. regular triangular pyramid, cube. dodecahedron b. The first two Platonic solids have more familiar names. What are they? c. Verify that uler s ormula is true for the first three Platonic solids = 6 + 2, = 2 + 2, = Multiple hoice A cube has a net with area 26 in. 2. ow long is an edge of the cube? A 6 in. in. 36 in. 4 in. Use uler s ormula to solve. 2. A polyhedron with 2 vertices and 30 edges has how many faces? A polyhedron with 2 octagonal faces and 8 rectangular faces has how many vertices? 6 4. escribe the cross section. raw each object. Then draw a horizontal and a vertical cross section. 38. a golf tee 39. a football 40. a baseball bat 4. hallenge 4. a banana 42. a pear 43. a bagel heck students work. 44. Writing ross 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. heck students work. 4. raw a solid that has the following cross sections. circle. raw and describe a cross section formed by a vertical plane cutting the left and back faces of a cube. heck students drawings; rectangle. horizontal vertical Visualization raw a plane intersecting a cube to get the cross section indicated. 46. scalene triangle 47. isosceles triangle 48. equilateral triangle 49. trapezoid 0. isosceles trapezoid. parallelogram Alternative Assessment ave each student bring a realworld polyhedron to class. ave them verify uler s ormula and then draw a net for the solid. 2. rhombus 3. pentagon 4. hexagon See margin. lesson quiz, PSchool.com, Web ode: aua-0 Lesson - Space igures and ross Sections

7 Test Prep Resources or additional practice with a variety of test item formats: Standardized Test Prep, p. 67 Test-Taking Strategies, p. 62 Test-Taking Strategies with Transparencies 9. [2] a. square b. Answers may vary. Sample: trapezoid Test Prep Multiple hoice Short Response Mixed Review or xercises 6, you may need uler s ormula, + V = A polyhedron has four vertices and six edges. ow many faces does it have? A. 2 B B 6. A polyhedron has three rectangular faces and two triangular faces. ow many vertices does it have? J The plane is horizontal. What best describes the shape of the cross section? A. rhombus B. trapezoid. parallelogram. square 8. The plane is vertical. What best describes the shape of the cross section? J. pentagon. square. rectangle J. triangle 9. raw 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. [] only correct drawing O for elp Lesson Probability A shuttle bus to an airport terminal leaves every 20 min from a remote parking lot. raw 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 bus to leave the parking lot. 60% ames A dartboard is a circle with a 2-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? 2% Lesson 0-3 ind the area of each equilateral triangle with the given measure. Leave answers in simplest radical form. Lesson side 2 ft 63. apothem 8 cm 64. radius 00 in. "3 ft 2 92"3 cm 2 700"3 in. 2 ind the value of x to the nearest tenth x x The lengths of the diagonals of a rhombus are 4 cm and 6 cm. ind the measures of the angles of the rhombus to the nearest degree. 67 and hapter Surface Area and Volume 604