Web Appendicies

8/9/2019 Web Appendicies

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W E B A P P E N D I X 1: A M E R I C A N N A T I O N A L S T A N D A R D SO F I N T E R E S T T O D E S I G N E R S ,A R C H I T E C T S , A N D D R A F T E R S

WEB SITESANSI . . . . . . . . . . . . . . . . . . . . . . . . . . .www.ansi.orgASME . . . . . . . . . . . . . . . . . . . . . . . . . .www.asme.org

TITLE OF STANDARDAbbreviation s for Use on Drawings and Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y1.11999American National Standard Drafting Practice:

Metric Drawing Sheet Size and Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.1M1995Decimal Inch Drawing Sheet Size and Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.11995Line Convention s and Lettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.2M1992 (R1998)Multi and Sectional View Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.3M1994 (R1999)Pictorial Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.41989 (R1999)Revision of Engineering Drawings and Associated Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.35M1992Dimension ing and Tolerancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5M1994(R1999)Dimension ing and Tolerancing with Mathemat ical Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5.1M1994 (R1999)Certification of Geometric Dimensioning and Tolerancing Professionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5.2(1995)Screw Thread Representation, Engineering Drawing and Related

Documentation Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.61978 (R1998)Engineering Drawing and Related Documentation Practices

Screw Thread Representation (Metric Supplemen t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.6AM1981 (R1998)Gears and Splines

Gear Drawing StandardsPart 1, for Spur, Helical, Double Helical, and Rack . . . . . . . . . . . . . . . . .ASME Y14.7.11971 (R1998)Gear and Spline Drawing StandardsPart 2, Bevel and Hypoid Gears . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.7.21978 (R1999)

Castings and Forgings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.8M1989 (R1996)Engineering Drawing and Related Documentation Practices

Mechanical Spring Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.13M1981 (R1998)Electrical and Electronics Diagrams (includes supplements ANSI Y14.15a1971

and ANSI Y14.15b1973) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.151966 (R1988)Fluid Power Systems and ProductsMoving Parts Fluid Con trols

Method of Diagramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME/(NFPA) T3.28.9R11989Engineer ing Drawings and Related Documen tation PracticesOptical Parts . . . . . . . . . . . . . . . . . . .ASME Y14.18M1986 (R1998)Types and Applications of Engineering Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.241999 (R2000)

Digital Representation for Communication of Product Definition Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .US PRO/IPO1001993Chassis FramesPassenger Car and Light Truck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.32.1M1999Parts Lists, Data Lists, and Index Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.34M1989Surface Texture Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.361978 (R1996)

Graphic Symbols:Electrical Wiring and Layout Diagrams Used in Architecture

and Buildin g Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.91972 (R1989)Plumbing Fixtures for Diagrams Used in Architecture and Building Construction . . . . . . . . . . . . . . . . . .ANSI Y32.41977 (R1999)Railroad Maps and Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.71972 (R1994)Fluid Power Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.101967 (R1999)Process Flow Diagrams in the Petroleum and Chemical Industries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.111961 (R1998)Mechanical and Acoustical Elements as Used in Schematic Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.181972 (R1998)Pipe Fittings, Valves, and Piping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/ASME Y32.2.31949 (R1999)Heating, Ventilating, and Air Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.2.41949 (R1998)Heat-Power Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.2.6M1950 (R1999)Welding, Brazing, and Nondestructive Examinat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/AWS A2.41993

Letter Symbols:Glossary of Terms Concern ing Letter Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.11972 (R1988)Quant ities Used in Electrical Science and Electrical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/IEEE 2801985 (R1992)Letter Symbols and Abbreviations for Quantities Used in Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/ASME Y10.111984Chemical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.121955 (R1988)Illuminating Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.181967 (R1977)Mathemat ical Signs and Symbols for Use in Physical Sciences and Technology . . . . . . . . . . . . . . . . . . . . . . .ANSI/IEEE 260.31993

Engineer ing Drawing Pract ices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.100M1998Engineer ing Drawings and Associated Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.35M1997

APPEN DIX 1 s


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W E B A P P E N D I X 2 : D I M E N S I O N I N G A N DT O L E R A N C I N G S Y M B O L S

2 s Ap pe ndix 2


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adhesive bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABDarc welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AW

atomic hydrogen welding . . . . . . . . . . . . . . . . . . . AHWbare metal arc welding . . . . . . . . . . . . . . . . . . . . BMAWcarbon arc welding . . . . . . . . . . . . . . . . . . . . . . . . CAW

gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAW-Gshielded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAW-Stwin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAW-T

electrogas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EGWflux cored arc welding . . . . . . . . . . . . . . . . . . . . . FCAWgas metal arc welding . . . . . . . . . . . . . . . . . . . . . GMAW

pulsed arc . . . . . . . . . . . . . . . . . . . . . . . . . . GMAW-Pshort circuiting arc. . . . . . . . . . . . . . . . . . . . GMAW-S

gas tungsten arc welding . . . . . . . . . . . . . . . . . . . GTAW

pulsed arc. . . . . . . . . . . . . . . . . . . . . . . . . . . GTAW-Pplasma arc welding. . . . . . . . . . . . . . . . . . . . . . . . . PAWshielded metal arc welding . . . . . . . . . . . . . . . . . SMAWstud arc welding. . . . . . . . . . . . . . . . . . . . . . . . . . . . SWsubmerged arc welding. . . . . . . . . . . . . . . . . . . . . . SAW

series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SAW-Sbrazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B

arc brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABblock brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BBcarbon arc brazing . . . . . . . . . . . . . . . . . . . . . . . . . CABdiffusion brazing. . . . . . . . . . . . . . . . . . . . . . . . . . . DFBdip brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DBflow brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FLBfurnace brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FB

induction brazing. . . . . . . . . . . . . . . . . . . . . . . . . . . . IBinfrared brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . IRBresistance brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . RBtorch brazing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TB

other welding processeselectron beam welding. . . . . . . . . . . . . . . . . . . . . . EBW

high vacuum . . . . . . . . . . . . . . . . . . . . . . . . EBW-HVmedium vacuum . . . . . . . . . . . . . . . . . . . . . EBW-MVnonvacuum . . . . . . . . . . . . . . . . . . . . . . . . . EBW-NV

electroslag welding. . . . . . . . . . . . . . . . . . . . . . . . . ESWflow welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . FLOWinduction welding. . . . . . . . . . . . . . . . . . . . . . . . . . . IWlaser beam welding . . . . . . . . . . . . . . . . . . . . . . . . LBWthermit welding. . . . . . . . . . . . . . . . . . . . . . . . . . . . TW

oxyfuel gas welding . . . . . . . . . . . . . . . . . . . . . . . . . OFWair acetylene welding . . . . . . . . . . . . . . . . . . . . . . . AAWoxyacetylene welding . . . . . . . . . . . . . . . . . . . . . . OAWoxyhydrogen welding . . . . . . . . . . . . . . . . . . . . . . OHWpressure gas welding . . . . . . . . . . . . . . . . . . . . . . . PGW

resistance welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . RWflash welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FWpercussion welding . . . . . . . . . . . . . . . . . . . . . . . . PEWprojection welding. . . . . . . . . . . . . . . . . . . . . . . . . . PWresistance seam welding. . . . . . . . . . . . . . . . . . . . RSEW

high frequency . . . . . . . . . . . . . . . . . . . . . . RSEW-HFinduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . RSEW-I

resistance spot welding . . . . . . . . . . . . . . . . . . . . . RSWupset welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UW

high frequency. . . . . . . . . . . . . . . . . . . . . . . . UW-HFinduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UW-I

soldering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sdip soldering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DSfurnace soldering. . . . . . . . . . . . . . . . . . . . . . . . . . . . FSinduction soldering . . . . . . . . . . . . . . . . . . . . . . . . . . ISinfrared soldering. . . . . . . . . . . . . . . . . . . . . . . . . . . IRSiron soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INSresistance soldering . . . . . . . . . . . . . . . . . . . . . . . . . . RStorch soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TSwave soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WS

solid-state welding . . . . . . . . . . . . . . . . . . . . . . . . . . . SSW

coextrusion welding . . . . . . . . . . . . . . . . . . . . . . . CEWcold welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CWdiffusion welding . . . . . . . . . . . . . . . . . . . . . . . . . DFWexplosion welding . . . . . . . . . . . . . . . . . . . . . . . . . EXWforge welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . FOWfriction welding . . . . . . . . . . . . . . . . . . . . . . . . . . . FRWhot pressure welding. . . . . . . . . . . . . . . . . . . . . . . HPWroll welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROWultrasonic welding. . . . . . . . . . . . . . . . . . . . . . . . . USW

thermal cutting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TCarc cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC

air carbon arc cutting . . . . . . . . . . . . . . . . . . . . . AACcarbon arc cutting . . . . . . . . . . . . . . . . . . . . . . . CACgas metal arc cutting . . . . . . . . . . . . . . . . . . . . GMACgas tungsten arc cutting . . . . . . . . . . . . . . . . . . GTACmetal arc cutting . . . . . . . . . . . . . . . . . . . . . . . . MACplasma arc cutting. . . . . . . . . . . . . . . . . . . . . . . . PACshielded metal arc cutting . . . . . . . . . . . . . . . . SMAC

electron beam cutting . . . . . . . . . . . . . . . . . . . . . . . EBClaser beam cutting. . . . . . . . . . . . . . . . . . . . . . . . . . LBC

air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-Aevaporative . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-EVinert gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-IGoxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-O

oxygen cut ting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OCchemical flux cutting . . . . . . . . . . . . . . . . . . . . . FOCmetal powder cutting . . . . . . . . . . . . . . . . . . . . . POCoxyfuel gas cutting . . . . . . . . . . . . . . . . . . . . . . . OFC

oxyacetylene cutting . . . . . . . . . . . . . . . . . . OFC-Aoxyhydrogen cutting. . . . . . . . . . . . . . . . . . OFC-Hoxynatural gas cutting . . . . . . . . . . . . . . . . OFC-Noxypropane cutting . . . . . . . . . . . . . . . . . . . OFC-P

oxygen arc cutting . . . . . . . . . . . . . . . . . . . . . . . AOCoxygen lance cutting . . . . . . . . . . . . . . . . . . . . . LOC

thermal spraying . . . . . . . . . . . . . . . . . . . . . . . . . . . THSParc spraying. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ASPflame spraying. . . . . . . . . . . . . . . . . . . . . . . . . . . . FLSPplasma spraying. . . . . . . . . . . . . . . . . . . . . . . . . . . . PSP

APPEN DIX 3 s

W E B A P P E N D I X 3 : D E S I G N A T I O N O F W E L D I N G A N DA L L I E D P R O C E S S E S B Y L E T T E R S

Welding and Letter Welding and LetterAllied Processes Designation Allied Processes Designation


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W E B A P P E N D I X 4 : S Y M B O L S F O R P I P E F I T T I N G SA N D V A L V E S

4 s A pp e ndi x 4


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Common Fractions

To add and subtract common fractions, the denominators (bot-toms) must be the same.

Normally, fractions are reduced to lowest form by dividingboth the numerator (top) and denominator by the same num-ber. Dividing by 4 reduces the previous fraction.

To multiply fractions, the denominators do not have to bethe same. Simply multiply the numerators and denominatorsseparately.

To divide fractions, flip the second fraction upside down(invert the divisor), then multiply.

When multiplying and dividing mixed numbers, it is neces-sary to convert them to fractions first. This is done by multi-plying the whole number by the denominator of the fractionand adding that to the top of the fraction for a new numerator.

For example:

The final answer was obtained by dividing the 99 by the 12with long division.

Decimal Fractions

To convert a common fraction to a decimal fraction, divide thenumerator by the denominator. Dividing 3 by 8 shows that

To convert a decimal to common fraction form, place thedecimal number without the decimal point on top and place itsplace value, given by this scheme, on the bottom of the fraction.

38

375=.

214

323

94

113

9912

814

= = =

638

6 8 38

518

= + =

14

23

14

32

38

= =

1

4

2

3

2

12

1

6 = =

416

14

=

1

163

164

16+ =

Examples are:

With the advent of computers and calculators, it has becomimportant to know how to round decimal answers properlyLook one place to the right of where the decimal is to brounded. If that digit is 5 or more, increase the value of thdigit to the left of it by one; if it is 4 or less, do not increase thleft digit. This concept is best grasped by viewing examples.

Rounding to th e nearest hun dredth:.275 becomes .28 (due to the 5 one place to the right o

the hundredth position).273 become .27 (the 3 is 4 or less).27499 also becomes .27 (the 9s h ave no effect on rou nd

ing to hundredths).279 becomes .28 (the 9 is 5 or more)

Rounding to the nearest tenth:.275 becomes .3.274 also becomes .3 (look only to the 7).249 becomes .2 (look only to the 4).05 becomes .1.04 becomes .0.729 becomes .7

Percentages

Percents are hundredths:

Percentages are easiest to calculate in decimal form. To convert a percentage to a decimal, move the decimal point twplaces to the left. Examples follow.

5% = .0550% = .503.75% = .0375

Success in working percentage problems requires correctlidentifying three pieces of the problem called the part, the baseand the rate. The part is a portion of any whole amount, tbase is the whole amount, and the rate is the number with percent sign (%).

To find the part:

1. Convert the rate to decimal form.

2. Multiply the rate by the base.

55

1001

20

5050

10012

%

%

= =

= =

.

.

.

375375

100038

01414

10007

500

6 75 675

1006

34

= =

= =

= =

APPEN DIX 5 s

W E B A P P E N D I X 5 : M A T H I N S T R U C T I O NA R I T H M E T I C

0 . 1 4 1 5 9 2tenthhundredththousandthten thousandthhundred thousandthmillionth


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Example 1: What is 15% of 233?In this example the rate is 15% and 233 is the whole amount

or base. The answer is 34.95 because 15% 233 = .15 233= 34.95.

Example 2: What is 5.25% of 1000?The answer is 52.5 because .0525 1000 = 52.5.

Example 3: What is 125% of 500?The answer is 625 because 1.25 500 = 625.

To find the base:

1. Convert the rate to decimal form.

2. Divide the part by the rate.

Example 4: 10% of what number is 375?This problem requires realizing that 375 is a portion of some

unknown whole amount. So you know the rate and part andare seeking the base.

The answer is 3,750 because 375 .10 = 3,750. Example 5: An old copy machine is known to waste 2% of thecopies it makes. How many copies were run if 14 were wasted?

The wasted copies are the part and the entire run is the base,so the answer is 14 .02 = 700 copies run.To find th e rate:

1. Divide the part by the base.

2. Convert the answer to percent by moving the decimalpoint two places to the right.

Example 6: What percent of 500 is 125?This type of problem requires identifying which number is

the base. From the wording, the whole amount is 500 so that isthe base, and 125 is a portion of this whole amount, making itthe part. Often, the base is the number immediately following

the word of. The answer to the problem is 25% because125 500 = .25, which equals 25%.

Example 7: A machine cutting tool has a useful life of 110 hours.If the tool is used 50 hours, what percentage of its useful life isleft?

In this problem first find the useful life left in hours:110 50 = 60. So this problem is the same as asking what per-cent of 110 is 60. The solution is 54.5% because 60 110 =.545 (after rounding), then .545 = 54.5%.

Example 8: $200 is what percent of $50?The answer is 400% because 200 50 = 4 = 400%. The

moral of this example is that the base is not always the larger

number.

Powers and Roots

A power is a small, raised number and stands for repeatedmultiplication.

Example 1: 53 = 5 5 5 = 125. This third power is called acube. The example is read, 5 cubed equals 125.

Example 2: 72 = 7 7 = 49. The second power is the square, andthis example is read, 7 squared equals 49.

A root is the reverse of a power.

Example 3: 3 125 = 5. This cube root of 125 equals 5 because5 cubed is 125.

Example 4: The square root is the most common root. Thesquare root of 49 may be written

2 49, but the index number 2is usually suppressed for square root. Write 49 = 7. It equals 7because 7 squared is 49.

Here are other examples of powers and roots.

2 5 = 32 6 2 = 36 3 8 = 2 16 = 4All scientific calculators have buttons or sequences of but-

tons that make finding powers and roots very easy. The book-let that comes with the calculator should be consulted to learnhow to do th is.

A L G E B R ASigned Numbers

The sign of a number is found immediately to the left of a num-ber. If there is no sign, this means the same as if there is a plussign. For example, in the expression 4 7 the 4 is a positivenumber and the 7 is a negative number. Multiplication of twonumbers is indicated by parentheses around one or both num-bers and no sign between the numbers: (3)(6) is multiplica-tion, but (3) +(6) is not multiplication. Here are other exam-ples of multiplication:

(2)(3) = 6 7(8) = 56

(5)(6) = 307(4) = 28

As these examples show, the rules for signs when multiply-ing are:

1. If signs are the same, the answer is positive.

2. If signs are different, the answer is negative.

Multiplication is also indicated by a raised dot (not to beconfused with a decimal point).

4 3 = 12The sign rules for division are identical to multiplication. For

example, 9 3 = 3. Often in algebra, division is indicated asa fraction. The previous division example might be expressed as

The rules for combining (adding and subtracting) signednumbers are:

1. If signs are the same, add the numbers and give the answerthe common sign.

2. If signs are different, subtract the two numbers and givethe answer the sign of the bigger.

Example 1: 1 +3This is not a multiplication problem because there are no

parentheses or raised dots; it is a combining problem. The 1 isnegative and the 3 is positive, so their signs are different, which

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The solution of larger simple equations involves applyingthese two basic operations more than once.

Example 9:

Example 10:

To solve an equation with a letter squared, take the squareroot of both sides of the equation.

Example 11:

Conversely, to solve an equation with the square root of aletter, square both sides of the equation.

Example 12:

Ratio and Proportion

One very common and useful type of equation is called a proportion. Each side of a proport ion is a fraction or ratio. To solvea proportion by cross-multiplying, multiply the top of one ratioby the bottom of the other and set them equal.

Example 1:

An older way to express this example is X:6::15:5. The x and5 were the extremes and the 6 and 15 the means.

Example 2: On a d rawing, the scale is 1:150. What actual lengthdoes a 3.25'' line of the drawing represent?

Set up a proportion.

115 0

3 25

1 150 3 25

487 5 40 7 1 2

' '' '

. ' '

.

. '' ' ' '

=

( ) = ( )( )=

xx

x or

x

x

xx

x

6155

5 6 15

5 9055

905

18

=

( )( )=( )( )=

=

=

y

y

y

=

( )==

8

8

64

22

because also equals7 7 49 ( )

x

xx and x

2

2

49

497 7

=

== =

= = + = =

=

2 9 112 11 92 2022

20210

yyyy

y

5 7 225 22 75 155

5

15

53

xxxx

x

+ == =

=

=

Example 3: A cylindrical container holds 1,000 gallons of oilwhen filled to a depth of 8 feet. How many gallons are therewhen the depth is 3 1 2 feet?

FIGURE 1 s A cylindrical container.

Set up a proportion.

G E O M E T R YTwo-Dimensional Figures

This section contains formulas for the perimeter (P) and areaof common geometric figures the drafter may encounter.Perimeter is the straight-line (linear) distance around a fig-ure, and area is the number of square units that fit within afigure.

Right Triangle

A + B = 90x2 +y2 = r 2P = x +y + r

FIGURE 2 s Right triangle.

Area12

xy=

10008

8 1000 3 5

8 3500437 5

.

.

gallonsfeet

x gallons3.5 feet

x

xx gallons

=

( )( )= ( )( )==

8 s Ap pe ndi x 5


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General TriangleA +B +C = 180P = a +b + c

RectangleP = 2a + 2bArea = ab

FIGURE 4 s Rectangle.

ParallelogramA +B = 180P = 2a + 2bArea = bh

FIGURE 5 s Parallelogram.

FIGURE 6 s Rhombus.

RhombusP = 4a

Note: The letters and p and q represent diagonal distances.p2 +q2 = 4a 2

Area12

pq=

d a b2 2= +

Area12

bh

Area s s a s b s c where s12

a b c

=

= ( ) ( ) ( ) = + +( )

CircleR = radiusD = diameter = 3.14159 . . .C = the circles perimeter, or circumferenceC = DArea = r2

Circle Inscribed with in a Right Triangle

FIGURE 8 s Circle within a right tr iangle.

Circle Inscribed within a General Triangle

FIGURE 9 s Circle within a general triangle.

Rs a s b s c

swhere s

12

a b c=( ) ( ) ( )

= + +( )

Rab

a b c=

+ +

AreaD4

2

=

APPEN DIX 5 s

FIGURE 3 s General tr iangle.

FIGURE 7 s Circle.


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Circle Circumscribed around a Right Triangle

Note: The letter c is the diameter of the circle as well as thehypotenuse of the right triangle.

FIGURE 10 s Circle around a right triangle.

Circle Circumscribed around a General Triangle

FIGURE 11 s Circle around a general triangle.

Angle Relationship for the Genera1 Triangle Inscribedwithin a Circle

FIGURE 12 s Angles within an inscribed triangle.

B1

2D=

Ra

2 sin Ab

2 sin Bc

2 sin C= = =

R12

c=Regular Polygonsn = number of sidesP = nf

FIGURE 13 s Regular polygon.

Arc of a Circles = length of the arc of the circle (arc length)Note: must be in radians.

FIGURE 14 s Arc of a circle.

Ellipse

(The perimeter formula is approximate.)Area = xy(The area formula is exact.)

FIGURE 15 s Ellipse.

P 2x y

2

2 2

= +

s = R

F 2R sin2

g R cos2

h R g

=

=

=

=

=

=

=

n 2n

180

f 2R sin180

nArea

12

nR sin360

n

df

2 tan180

n

2

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Three-Dimensional Figures

This section shows formulas for total surface area (S) in squareunits and volume (V) in cubic units for common shapes.

Rectangular SolidS = 2(wh + lw + lh)V = lwh

FIGURE 16 s Rectangular solid.

CylinderS = 2R2 + 2RhV = R2h

FIGURE 17 s Cylinder.

Right Circular Cone

FIGURE 18 s Right circular cone.

Sphere

FIGURE 19 s Sphere.

S RS D where D = 2R

V43

R

2

3

===

4 2

,

S R RL

L R h

V 13

R h

2

2 2

2

= += +

=

T R I G O N O M E T R Y Right Triangle Trigonometry

Trigonometry is based upon the lengths of sides and the angleof right triangles. It is best to treat the trigonometric functidefinitions as simply working formulas involving two sides a

an angle of the right triangle of Figure 20 and to use the formula that involves the information in the problem. Here are thdefinitions of the trig functions: In the right triangle, thlongest side (r) is called the hypotenuse. The side (y) is calthe opposite side, because it is opposite to angle (A). Side (x) called the adjacent side.

The first formula says that dividing the length of side y bhypotenuse r gives a number called the sine of angle A. (Sinabbreviated sin but still pronounced sine. )

FIGURE 20 s Right tr iangle used for trigonometry definitions.

Example 1: A very common right triangle is the 3-4-5 right triangle. In Figure 21, find the sine of angle A.

Using the sine formula:

FIGURE 21 s 3-4-5 right triangle.

Sin Ayr

Sin A35

=

= =.6

Sin Ayr

Cos Axr

Tan Ayx

=

=

=

Sin Ayr

Cos Axr

Tan Ayx

=

=

=

APPEN DIX 5 s


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The usefulness of trigonometry is that it provides a connec-tion between the lengths of sides and measurement of angles.Knowing, for example, that sin A = .6, a calculator can be usedto find angle A. For most calculators, entering .6 then pushingINV and SIN (or 2ND and SIN) gives angle A as 36.9.

Similarly, cosine is the adjacent side divided by thehypotenuse, and tangent is the opposite side divided by the

adjacent side. An additional formula used to solve right trian-gles involves only the sides. It is the Pythagorean theorem:

x2 + y 2 = r 2

Example 2: Find the diagonal distance in Figure 22 for a rec-tangular plate. Using the formula:

x2 + y2 = r2

72 + 10 2 = r2

49 + 100 = r 2

149 = r 2

FIGURE 22 s Rectangular plate.

Example 3: Find th e length of side x for the right triangle in Fig-ure 23.

Because you know angle A and side y, use the tangent for-mula; only tangent involves the unknown side x and theknown angle and side.

FIGURE 23 s Right triangle.

Angle Conversion and Arc Length

One degree equals 60 minutes of arc: 1 = 60'One minute equals 60 seconds of arc: 1' = 60''

Therefore, 1 = 3600''

Example 1: Convert 450'35'' to decimal degrees (nearesthundredth).

tan A = yx

tan 4012''

x

.839112''

xx

x12''

.8391x 14.3''

=

=

=

=

=

. ''8391 12

r = =149 12 21. ' '

The answer is

Example 2: Convert 10.268 to degrees, minutes, and seconds.Working with the decimal fraction of the degrees, .268 60

= 16.08 for 16 whole minutes. Then, working with the decimalfraction of minutes, .08 60 = 4.8 or (rounding off) 5 wholeseconds. The complete answer becomes 1016'5''.

Here are additional conversion facts:

Degrees and Radians:

180= radians (where = 3.14159 . . .)1 radian = about 57.3

1= about .01745 radian

Example 3: Convert 30 to radians (nearest thousandth). Theanswer is 30 .01745 = .5235 or .524 radian.

Example 4: Convert 2 radians to degrees (nearest tenth). Theanswer is 2 57.3 = 114.6.

Vectors

A vector is a directed line segment, or arrow, with two attrib-utes: length (magnitude) and angle (direction). Points in theplane may be specified by their x and y coordinates or by mag-nitude and direction of a vector with its tail at the origin and itshead at the point in the plane as shown in Figure 24.

Vector notation and equations conform to the definitions of the trig functions except that Greek letter (theta) is oftenused for the angle A. Here are the conversion formulas:

Polar to Rectangular (vector form to x-y form):

x = r cos y = r sin

The x and y are also called vector components.

FIGURE 24 s Vector.

45060

353600

4 8333 0097

4 843

+ + = + +=

. .

. or 4 . 8 4

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Example 1: Find the components of the vector shown in Figure 25.Using the conversion formulas with r = 10 and = 30,

x = (10)(cos 30 ) = (10 )(.866 0) = 8.66 lb, andy = (10)(sin 30) = (10)(.5) = 5 lb

FIGURE 25 s Vector in polar form.

Rectangular to Polar (x-y form to vector form):

Example 2: Convert the coordinates (7,13) in Figure 26 to polarform.

Using the conversion formulas with x = 7 and y = 13,

FIGURE 26 s Vector in rectangular form.

Slope of a Line

The slope-intercept form of the equation of a straight line is:y = mx + b. Letter m is the slope (rise divided by run of the

r 7 1

Inv Tan137

Inv Tan 1.857 = 61.7

2 2= + = + = =

= =

3 49 169 218 14 8.

r x y

Inv Tanyx

2 2= +

=

line), and letter b is the y-axis intercept (the point on the vertcal axis that the line crosses). For example, Figure 27 is thgraph of the equation y = 2x + 5.

FIGURE 27 s Graph of y = 2x +5.

Example 1: What is the equation of the line in Figure 28?The slope is 12 3 = 4 and the y-intercept is -5, so the equ

tion is y = 4x 5.Positive slopes in math slant up and to the right; negativ

slopes slant down and to the right.

FIGURE 28 s Graph requiring an equation.

Example 2: What is the equation of the line in Figure 29?The slope is 6 3 = 2 and the y-intercept is 4, so the equ

tion is y = 2x +4.

FIGURE 29 s Graph requiring an equation.

APPEN DIX 5 s


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Oblique Triangles

In trig, lowercase letters usually stand for the lengths of sidesand capital letters stand for angles, as shown in Figure 30.

FIGURE 30 s General oblique tr iangle.

To solve a triangle means to find all missing angles andsides. The Law of Sines can be used to solve a general trianglewhen one side and the opposite angle are known. The Law of Sines is usually expressed as:

It is really three separate equations; each one is a proportion .

Example 1: Find side b in the triangle shown in Figure 31.Angle B can be found quickly knowing that the sum of the

angles of any triangle is 180.

B = 180 (80 + 45) = 180 125 = 55Now the Law of Sines can be used:

FIGURE 31 s Oblique triangle with unknown side.

aSin A

bSin B

14Sin 45

bSin 55

14.7071

b.8192

.7071b .8192 14

.707 1b 11 .4 86b 16.2' '

=

= =

=( )( )==

aSin A

bSin B

bSin B

cSin C

cSin C

aSin A

=

=

=

a

Sin Ab

Sin Bc

Sin C= =

When an oblique tr iangle has an angle greater than 90, caremust be taken when using the Law of Sines to find that angle.The calculator gives only angles less than 90 when INV SIN ispushed. A theorem from trig class must be used: sin = sin(180 ), which means the calculator answer must be sub-tracted from 180 to get the true answer. The next exampleillustrates this.

Example 2: Find angle A in Figure 32. First, convert to inches:16' 8'' = 200'' and 14' 2'' = 170'', then using the version of the Law of Sines involving as and cs:

FIGURE 32 s Oblique triangle with unknown angle.

Angle A is known to be greater than 90, so A = 180 17.7 = 162.3. This final subtraction from 180 is taken onlywhen finding an angle greater than 90 with the Law of Sines.

Sometimes a side and opposite angle are not available. Inthat case, the triangle may be solved with the Law of Cosines.Again, there are three versions of this law.

a 2 = b 2 + c2 2bc cos Ab 2 = a 2 + c2 2ac cos Bc2 = a 2 + b 2 2ab cos C

aSin A

cSin C

20 0Sin A

17 0Sin 15

20 0Sin A

17 0.2588

Sin A 200Sin A 51.76

170 Sin A

Sin A .3045A Inv Sin .3045A 17.7

=

= =

=( )( )=

====

170 258817 0

17051 76170

.

.

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Example 3: Find side b of the triangle in Figure 33.Use the version of the Law of Cosines containing angle B.

FIGURE 33 s Oblique triangle with unknown side.

b 14 2 11 14 cos 130

b

b

bb 514.98b 22.7' '

2 2

2

2

2

= + ( )( ) ( )= + ( )( )( )= + +===

11

121 196 2 11 14 6428

121 196 197 98

5 14 9 8

2

.

.

.

Example 4: Find angle A of the triangle in Figure 34.Use the version of the Law of Cosines containing angle A.

FIGURE 34 s Oblique triangle with unknown angle.

6 8 2 100 cos A

3 cos A3 cos A

3,600 17, 569 17, 400 cos A

13, 969 17, 400 cos A17,400 cos A

17,400cos A

A Inv cos .8028A 36.6

2 20 100 7 87

600 10 000 7 569 17 400600 17 569 17 400

13 96917 400

8028

2= + ( )( )= + =

= = =

===

, , , ,, , ,

,,

.

APPEN DIX 5 s

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