Packages and Polygons

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D Polyhedra Euler discovered his formula around 1750, but it was not until 1794 that a French mathematician named Adrien-Marie Legendre proved it actually worked for all polyhedra. The following is one way of explaining why Euler’s formula works for all polyhedra. Tetrahedron Begin with any tetrahedron: It has four faces, four vertices, and six edges. Euler’s formula states that: F V E 2 Verifying Euler’s formula: 4 4 6 2 Step 1: Cut off one tip of the tetrahedron. Step 1 This changes the number of: • faces: F increases by 1. • vertices: V increases by 2. • edges: E increases by 3. 12. a. Explain the changes in Step 1. Step 2 b. Why doesn’t the formula F V E change in Step 1? Step 2: Cut off a different tip of the tetrahedron. 13. a. Describe the changes in the number of faces, vertices, and edges. Step 3 b. Do the changes affect Euler’s formula? Why or why not? 14. Describe Steps 3 and 4. Discuss any changes. 15. a. Describe the semi-regular polyhedra produced after Step 4. Step 4 b. Verify Euler’s formula for this solid. 38 Packages and Polygons
Polyhedra D When you cut a piece off each vertex of an icosahedron, you get a semi-regular polyhedron made of pentagons and hexagons. 16. Here you have five semi-regular polyhedra. Match each semiregular polyhedron to its original Platonic solid. Section D: Polyhedra 39
Page 1 and 2: Packages and Polygons Geometry and
Page 3 and 4: The Mathematics in Context Developm
Page 5 and 6: Contents Letter to the Student vi S
Page 7 and 8: A Packages Sorting Packages José d
Page 9 and 10: Packages A 4. Use the distinguishin
Page 11 and 12: Packages A 6. What shape would each
Page 13 and 14: Packages A 9. Explain what would ha
Page 15 and 16: Packages A Ne(a)t Problems 15. a. D
Page 17 and 18: Faces Each flat side of a shape is
Page 19 and 20: 3. Draw a net of a rectangular pris
Page 21 and 22: Bar Models B Harold starts to make
Page 23 and 24: Bar Models B To make a frame rigid,
Page 25 and 26: Bar Models B Math History Mankind h
Page 27 and 28: Sylvia used graph paper to make a d
Page 29 and 30: C Polygons Put a Lid on It Susanne
Page 31 and 32: Polygons C a c b d Susanne must fin
Page 33 and 34: Polygons C Angles You can find the
Page 35 and 36: 1. a. Draw two different shapes tha
Page 37 and 38: D Polyhedra A polyhedron is a three
Page 39 and 40: Polyhedra D You can make bar models
Page 41 and 42: Polyhedra D The simplest Platonic s
Page 43: Polyhedra D Semi-Regular Polyhedra
Page 47 and 48: Semi-Regular Polyhedron Semi-regula
Page 49 and 50: E Volume Candles In the previous se
Page 51 and 52: Volume E Rosa and Lydia decide to b
Page 53 and 54: Volume E From the unit Reallotment,
Page 55 and 56: Volume AE Formulas for Volume In th
Page 57 and 58: The slices are not identical for py
Page 59 and 60: Additional Practice Section A Packa
Page 61 and 62: Additional Practice Kim has a piece
Page 63 and 64: Additional Practice This shape is a
Page 65 and 66: Section A Packages 1. Descriptions
Page 67 and 68: Answers to Check Your Work 3. He ca
Page 69 and 70: Answers to Check Your Work 4. a. Th
Page 71 and 72: Answers to Check Your Work Section

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