In this post, we will see the book *Geometry* by E. Z. Shuvalova.

# About the book

The book provides a comprehensive treatment of basic geometrical concepts for high schools. The first chapter deals with properties of triangles. Chapter 2 gives a comprehensive introduction to structure of solid geometry. The chapter on parallelism in space details the concept in various contexts. Next three chapters dal with space, vectors and planes and their properties including dihedral and polyhedral angles. The last few chapters treat solids, polyhedrons and their properties such as areas volumes and sections by planes. There are worked problems and exercises for each chapter.

The book was translated from Russian by Leonid Levant and was published in 1980 by Mir.

Credits to the original uploader.

You can get the book here.

Follow us on The Internet Archive: https://archive.org/details/@mirtitles

Follow Us On Twitter: https://twitter.com/MirTitles

Write to us: mirtitles@gmail.com

Fork us at GitLab: https://gitlab.com/mirtitles/

Add new entries to the detailed book catalog here.

# Contents

Symbols Used in the Book 9

**CHAPTER 1. BASIC RELATIONS BETWEEN THE ELEMENTS OF AN ARBITRARY TRIANGLE. SOLVING OBLIQUE TRIANGLES**

Sec. 1. The Law of Sines 11

Sec. 2. The Law of Cosines 13

Sec. 3. Expressing the Tangent of a Half-Angle in Terms of the Sides of a Triangle and Radius of the Inscribed Circle (or the Incircle) 16

Sec. 4. Solving Oblique Triangles 17

Sec. 5. Measuring Distances Between “Inaccessible” 20

Sec. 6. Other Problems on Solving Oblique Triangles 23

Sec. 7. Worked Problems 28

Problems to Chapter 4 32

**CHAPTER 2. LOGICAL STRUCTURE OF THE COURSE OF SOLID **

**GEOMETRY**

Sec. 8. Structure of the Course of Solid Geometry. Notation and Terminology 36

Sec. 9. Axioms of Belonging 38

Sec. 10. Axioms of Distance 40

Sec. 11. Axioms of Order 42

Sec. 12. Axiom of Plane Mobility 45

Sec. 13. Axiom of Parallelism 47

Sec. 14. Corollaries Following from the Axioms 48

**CHAPTER 3. PARALLELISM IN SPACE**

Sec. 15. The Relative Position of Two Straight Lines in

Space. The Relative Position of a Plane and a Straight Line in Space 50

Sec. 16. Parallelism of a Straight Line and a Plane 51

Sec. 17. Parallel Planes 52

Sec. 18. Direction in Space. The Angle Between Two Lines 55

Sec. 19. Parallel Projecting 58

Sec. 20. Representation of Figures in Solid Geometry 61

Sec. 21. Worked Problems 63

Problems to Chapter 3 66

**CHAPTER 4. TRANSFORMATION OF SPACE. VECTORS**

Sec. 22. Transformation of Space 68

Sec. 23. Translation in Space 69

Sec. 24. The Vector Defined 70

Sec. 25. The Sum of Vectors 72

Sec. 26. Subtraction of Vectors. Multiplication of a Vector by a Number 75

Sec. 27. A Linear Combination of Vectors. The Conditions of Collinearity and Coplanarity 78

Sec. 28. Scalar Product of Vectors 81

Sec. 29. Arithmetic Properties of a Scalar Product 83

Sec. 30. Vector Product of Vectors 85

Problems to Chapter 4

**CHAPTER 5. PERPENDICULARITY IN SPACE. DIHEDRAL AND POLYHEDRAL ANGLES**

Sec. 31. A Perpendicular to a Plane 91

Sec. 32. An Inclined Line and Its Projection on a Plane. The Distance from a Point toa Plane 92

Sec. 33. The Theorem on Three Perpendiculars 94

Sec. 34. The Angle Between an Inclined Line and a Plane 97

Sec. 35. Dependence Between Parallelism and Perpendicularity of Straight Lines and Planes 98

Sec. 36. The Distance Between Skew Lines 101

Sec. 37. The Triple Product of Three Vectors. The Test for Coplanarity of Three Vectors 102

Sec. 38. Dihedral Angles 104

Sec. 39. The Angle Between Planes. Perpendicular Planes 106

Sec. 40. Orthogonal Projecting 108

Sec. 41. The Length of a Projection of a Line Segment. The Area of a Projection of a Plane Polygon 110

Sec. 42. The Area of a Projection of an Arbitrary Plane 113

Sec. 43. Polyhedral Angles 114

Sec. 44. Worked Problems 118

Problems to Chapter 5 123

**CHAPTER 6. THE COORDINATE METHOD**

Sec. 45. Rectangular Coordinate System 127

Sec. 46. Expressing a Scalar Product of Vectors in Terms of Their Coordinates. The Equation of a Plane 132

Sec. 47. Expressing the Vector Product of Two Vectors in Terms of Their Coordinates 135

Sec. 48. Expressing the Triple Product of Three Vectors in Terms of Their Coordinates 138

Sec. 49. Worked Problems 138

Problems to Chapter 6. 145

**CHAPTER 7. POLYHEDRA, CYLINDERS, CONES**

Sec. 50. The Polyhedron Defined 147

Sec. 51. Regular Polyhedra 147

Sec. 52. Eulers Theorem 149

Sec. 53. The Prism. 152

Sec. 54. A Cylindrical Surface. The Cylinder 156

Sec. 55. The Pyramid 159

Sec. 56. A Conical Surface. The Cone 160

Sec. 57. Homothety in бресе 163

Sec. 58. Properties of Parallel Sections of a Cone (Pyramid). Frustums of a Cone (Pyramid) 165

Sec. 59. Sections of Polyhedra 167

Sec. 60. Worked Problems 169

Problems to Chapter 7 174

**CHAPTER 8. THE BALL**

Sec. 61. A Sphere and a Ball Defined. A Sphere and a Ball Cut by a Plane 181

Sec. 62. A Tangent Plane. 182

Sec. 63. The Concept of a Spherical Triangle 183

Sec. 64. Worked Problems 485

Problems to Chapter 8. 189

**CHAPTER 9. MEASURING VOLUMES**

Sec. 65. General Properties of Volumes 191

Sec. 66. The Volume of a Rectangular Parallelepiped 192

Sec. 67. The Volume of a Right Cylindrical Solid 194

Sec. 68. The Volume of an Oblique Cylindrical Solid 195

Sec. 69. The General Formula for Computing the Volume of a Figure Using the Areas of Cross-Sections 196

Sec. 70. The Formulas for Computing the Volume of a Cone, a Ball and Its Parts. Simpson’s Formula 199

Sec. 71. Worked Problems 204

Problems to Chapter 9 209

**CHAPTER 10. THE AREA OF SURFACE**

Sec. 72. The Area of the Surface of a Polyhedron 215

Sec. 73. The Area of an Arbitrary Surface 218

Sec. 74. The Area of the Surface of a Right Circular Cylinder, a Circular Cone, and a Ball 219

Sec. 75. Worked Problems 223

Problems to Chapter 10 226

ANSWERS 232

INDEX 236

Pingback: Geometry – Shuvalova — Mir Books | Chet Aero Marine