Frequently Used Formulas All Triangles P = a + b + c B = 1 A 2bh c h a = 1 A C S 2P b A = 2s(s - a)(s - b)(s - c) (Hero’s formula) Right Triangles c2 = a2 + b2 B 45° : 45° : 90° :: 1 : 1 : 12 30° : 60° : 90° :: 1 : 13 : 2 ca 3 : 4 : 5 (ratio of sides) AC 5 : 12 : 13 (ratio of sides) b Polygons b Rectangle Trapezoid w ah c A = l * w A = 1 h(B + b) l 2 B P = 2l + 2w P = a + b + c + B Square Regular Octagon 135° 2 s F = = D A s P 8s S P = 4s A = 0.7071 D2 or 2 Parallelogram b A = 0.8284 F A = b * h haor P = 2a + 2b A = 4.8275 S2 N-sided Polygons Any polygon Regular polygon only Degreestotal = 180° (N - 2) Degreeseach angle = 180° (N - 2)>N Circles C = 2pr or d C = pd r A = pr2 Volumes and Surface Areas Prism Cylinders Base 2 LSA = lateral surface area V = pr h = p TSA = total surface area LSA 2 rh Height B = area of base or r V = B * h LSA = pdh 2 LSA = area of all faces TSA = LSA + 2pr TSA = LSA + 2B Cones 1 = p 2 h s Face V 3 r h h LSA = prs r (s = slant height) Base Temperature Conversions C = 5>9(F - 32°) (Fahrenheit to Celsius) F = 9>5C + 32° (Celsius to Fahrenheit) Right Triangle Trigonometry Ratios Sin u = opp/hyp hyp Cos u = adj>hyp opp ␪ Tan u = opp>adj adj Mathematics for Carpentry and the Construction Trades THIRD EDITION Alfred P. Webster Kathryn E. Bright Prentice Hall Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editorial Director: Vernon R. 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Catalogue in Publication data available from the Library of Congress 10987654321 ISBN 10: 0-13-511400-4 ISBN 13: 978-0-13-511400-1 Preface This text is intended to meet the needs of a two-semester course for students of carpentry and building construc- tion. Throughout, the emphasis has been on simplicity. Illustrated problems include step-by-step explanations of their solutions. Explanations have been presented in summary form with the necessary steps highlighted for easy identification. Drawings and photographs have been included to clarify the particular subject under discussion. In the interest of drawing the reader’s attention to facts that affect our environment and remarks that provoke ways of making the greening of our world more of a conscious effort, at the beginning of each chapter is a “Did You Know” statement. Chapters 1 through 10 cover the fundamental mathematics necessary to a broad range of skills. Although problems in these chapters apply to a variety of areas, the emphasis has been given to applications in the building construc- tion field. We suggest that calculators not be used in the first sections in Chapters 1 through 3 to allow students to increase their mathematical skills in some basic areas. Thereafter, calculator usage is encouraged, with emphasis on efficiency and accuracy. Chapters 11 through 27 cover matters of direct concern to the builder. The sequence of topics in these chapters follows the logical construction process insofar as is practical. Phases of construction normally relegated to sub- contractors (including masonry, plumbing, heating, and electrical, among others) have not been covered. The oc- currence of these phases of construction is of concern to the primary contractor, and their sequencing is alluded to in a summary chapter; however, we have made no attempt to include mathematics related to these areas. Although this text is not intended to be a complete “how-to” manual with respect to building techniques, a cer- tain amount of instruction has been included. In many areas of building, an understanding of the relevant mathe- matics is coupled with an understanding of how the construction is done. Furthermore, efficiency and accuracy (both highly desirable goals for the builder and estimator) are best achieved when an understanding of building methods has been reached. Users of this text will find the topic sequence logical and explanations clear and concise. The problems are re- alistic and practical and typical of the types of calculations that builders can expect to encounter in practice. An- swers to the odd-numbered exercises have been included in the back of the book. In addition, a solutions manual is available that contains many detailed solutions. This text is the result of the depth of experience we bring to users of this material. Our extensive backgrounds as teachers of applied mathematics and practitioners in the building construction field should make this book valu- able to its users. We wish to thank the reviewers of this edition for their helpful comments: Stephen Csehoski, Pennsylvania High- lands Community College; Barbara Ries, Chippewa Valley Technical College; Joseph Ryan, Cape Cod Community College, Eastern Maine Community College. Finally, we wish to express our appreciation to Sally Webster and Daisy Bright, both accomplished teachers of mathematics. Their identification of errors and suggestions for changes have been a valuable contribution to this effort. A. P. Webster K. E. Bright iii Contents 1 Whole Numbers 1 6 Percents 112 1.1 Place Value and Rounding 1 6.1 Conversion of Fractions and Decimals 112 1.2 Accuracy and Precision 4 6.2 Percent Problems 117 1.3 Order of Operations 7 6.3 Construction Applications Involving Percent 123 2 Fractions 13 2.1 Equivalent Fractions 14 7 Angles and Triangles 128 2.2 Finding the Lowest Common Denominator 7.1 Angle Measure 128 and Equivalent Fractions 24 7.2 Congruent and Similar Triangles 133 2.3 Addition of Fractions and Mixed 7.3 The Pythagorean Theorem 139 Numbers 29 7.4 Special Right Triangles 144 2.4 Subtraction of Fractions and Mixed 7.5 Perimeters and Areas of Triangles 148 Numbers 35 2.5 Multiplication of Fractions and Mixed 8 Areas and Perimeters 157 Numbers 41 2.6 Division of Fractions and Mixed 8.1 Quadrilaterals 157 Numbers 44 8.2 Circles 166 8.3 Octagons 178 3 Decimal Fractions 51 8.4 Composite Figures 180 3.1 Place Value and Rounding 52 3.2 Determining Precision of Decimal 9 Volume and Surface Area Fractions 55 of Solids 191 3.3 Addition of Decimals 56 9.1 Rectangular Solids and Prisms 191 3.4 Subtraction of Decimals 59 9.2 Cylinders and Cones 198 3.5 Multiplication of Decimals 61 9.3 Spheres 203 3.6 Division of Decimals 63 9.4 Composite Shapes 205 3.7 Multiplying and Dividing by Powers of 10 65 10 The Metric System 210 3.8 Decimal and Fraction Conversions 67 10.1 Metric–Metric Conversions 210 4 Weights, Measures, 10.2 Metric–American Conversions 217 and Conversions 77 10.3 Temperature Conversions 222 4.1 Linear Measure 77 4.2 Operations with Mixed Units 82 11 Board Measure 225 4.3 Unit Conversions 87 11.1 Finding the Number of Board Feet 225 4.4 Area and Volume Conversions 90 11.2 Finding Linear Feet When the Number of 4.5 Rate Conversion Problems 95 Board Feet Is Known 227 5 Ratio and Proportion 100 12 Lumber Pricing 231 5.1 Ratio 100 12.1 Pricing Systems 231 5.2 Proportion 104 12.2 Price Conversions 232 iv Contents v 13 Footings, Foundations, 20 Roofs III: Valley Rafters—The and Slabs 236 Conventional Case 295 13.1 Calculating the Amount of Components in a 20.1 Types of Valley Rafters 295 Concrete Mixture 236 20.2 Calculating the Length of a Conventional 13.2 Calculating the Amount of Concrete Valley Rafter 296 Needed 237 20.3 End Cuts 297 13.3 Foundations 238 13.4 Footings 240 21 Roofs IV: Jack Rafters 299 13.5 Slabs 241 21.1 Calculating Jack Rafter Decrease 299 14 Girders, Sill Plates, Bridging, Floor 21.2 The Framing Square 302 Joists, and Floor Covering 243 14.1 The Girder 243 22 Roofs V: Hip and Valley Rafters— 14.2 Sill Plates 245 The Unconventional Cases 304 14.3 Floor Joists 246 22.1 Calculating Unconventional 14.4 Bridging 249 Rafters 304 14.5 Floor Covering 250