Mathematics for Carpentry and the Construction Trades, 3rd Edition

413 Pages·2011·4.39 MB·English

by  Alfred P. Webster

Frequently Used Formulas
All Triangles
P = a + b + c
B
A = 1bh
2 c h a
S = 1P A C
2 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
c a
3 : 4 : 5 (ratio of sides)
5 : 12 : 13 (ratio of sides) A C
b
Polygons
b
Rectangle Trapezoid
w a h c
A = l * w A = 1h(B + b)
2
l B
P = 2l + 2w P = a + b + c + B
Square Regular Octagon
135°
A = s2 s P = 8s F
D
S
P = 4s A = 0.7071 D2
or
Parallelogram A = 0.8284 F2
b
A = b * h h a or
P = 2a + 2b A = 4.8275 S2
N-sided Polygons
Any polygon Regular polygon only
Degrees = 180° (N - 2) Degrees = 180° (N - 2)>N
total each angle
Circles
C = 2pr
or
d
C = pd r
A = pr2
Volumes and Surface Areas
Prism Cylinders Base
LSA = lateral surface area V = pr2h
TSA = total surface area LSA = 2prh Height
B = area of base or
r
V = B * h LSA = pdh
LSA = area of all faces TSA = LSA + 2pr2
TSA = LSA + 2B
Cones
V = 1pr2h h s
Face 3
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
(cid:2)
Tan u = opp>adj
adj
Mathematics for Carpentry
and the
Construction Trades
T H I R D E D I T I O N
Alfred P.Webster
Kathryn E. Bright
Prentice Hall
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Catalogue in Publication data available from the Library of Congress
10 9 8 7 6 5 4 3 2 1
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
7 Angles and Triangles 128
2.1 Equivalent Fractions 14
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
11 Board Measure 225
4.2 Operations with Mixed Units 82
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
22 Roofs V: Hip and Valley Rafters—
14.1 The Girder 243
The Unconventional Cases 304
14.2 Sill Plates 245
14.3 Floor Joists 246 22.1 Calculating Unconventional
14.4 Bridging 249 Rafters 304
14.5 Floor Covering 250
23 Stairs 309
15 Wall Framing 253
23.1 Determining Number of Risers (NRi) and
15.1 Shoes and Plates—Exterior Walls 254 Unit Rise (URi) 311
15.2 Exterior Wall Studs 254 23.2 Determining the Stairwell Opening
15.3 Interior Wall Framing 258 (SO) 313
15.4 Headers 259 23.3 Determining the Total Run (TRu) 314
23.4 Space Limitations 316
16 Roofs I: Common Rafters 262
16.1 The Slope Triangle 263 24 Framing and Covering Gable Ends;
16.2 Rafter Dimensions 265 Exterior Trim 319
16.3 Rafter Calculations 265
24.1 Framing Gable Ends 319
16.4 Overhangs 267
24.2 Sheathing and Siding Gable Ends 321
16.5 Applying the Numbers; Adjustments 269
24.3 Determining Materials for Exterior
16.6 Determining Slope of an Existing Roof 270
Trim 324
16.7 Determining the Angle of Elevation of a
Roof 272
25 Wall and Roof Covering 327
17 Introduction to the Framing
25.1 Estimating Exterior Sheathing 328
Square 276
25.2 Siding 330
25.3 Drywall 331
18 Overhangs 279
25.4 Roof Covering 332
18.1 Purposes of Overhang 279
18.2 Calculating Overhang and Materials 280 26 Heat Loss, Return on Investment,
Operating Costs 338
19 Roofs II: Hip Rafters—The
Conventional Case 287 26.1 Money Matters 338
26.2 Heat Loss 340
19.1 Calculating Hip Rafter Length (HRL)—
26.3 Operating Costs 349
Method 1 287
19.2 Calculating Hip Rafter Length (HRL)—
27 The Estimating Process 351
Method 2 290
vi Contents
Appendices 364 Answers to Odd-Numbered
Exercises 384
Appendix A Right-Triangle Trigonometry 364
Appendix B Resistances (R-Factors) of Various
Materials 377 Bibliography 400
Appendix C House Plans 378
Index 401
Glossary 382
1
C H A P T E R
Whole Numbers
Did You Know
Energy-efficient windows (double, triple, and/or low E glass) can save
30% or more in residential heating and cooling costs compared to
single-pane windows.
O B J E C T I V E S
Upon completing this chapter, the student will be able to:
1. Determine place value of numerals.
2. Round whole numbers to a given place value.
3. Distinguish between precision and accuracy (optional).
4. Set up problems involving more than one type of arithmetic operation.
5. Solve problems using the proper order of operations.
6. Add, subtract, multiply, and divide whole numbers and decimal numbers.
1.1 Place Value and Rounding
Place Value for Whole Numbers
To make sure that we understand the decimal system (the system we use), a brief review
of place value is in order. A numeral such as 562 means 500 (cid:2) 60 (cid:2) 2.In other words,
there are 5 hundreds, 6 tens, and 2 ones. Figure 1–1 indicates the place values for five
hundred sixty-two (562), one thousand eight hundred fifty-three (1853), and five million,
thirty thousand, four hundred six (5,030,406).
Figure 1–1
1