STRUCTURES AND SYSTEMS 2
Chapter 10: Measurement and Geometry
The two primary measurement systems used in the world today are the metric system, also known as the Système International d’Unités and commonly abbreviated to SI in all languages, and the U.S. customary units system, referred to in the United States as English units or standard units. The latter is an irregular system based on imperial units once used in the United Kingdom and the British Commonwealth.
The metric system has become the universally accepted system of units in science, trade, and commerce. In the United States, however, where the Metric Conversion Act of 1975 established SI as the preferred system of weights and measures for trade and commerce, federal laws have yet to mandate SI as the oficial system, making its use still primarily voluntary. Several U.S. agencies, including the American National Metric Council (ANMC) and the United States Metric Association (USMA), are working to estab-lish SI as the oficial measurement system, a process known as metrication. Though the architectural, engineering, and building trades have been slow to make a full transi-tion, nearly all federally funded building projects are now required to be in SI units.
UNITS OF MEASURE: CUSTOMARY UNIT DATA
Customary units may be shown in a number of ways, including as fractions (11/2”) or as decimals (1.5” or 0.125’), depending on the more common usage for a particular situation. It should be noted that, though not the case here, exponents can be used with abbreviations that designate area or volume; for example, 100 ft.2 for area or 100 ft.3 for volume.
Customary Unit Relation to Other of Measure Customary Units
inch (in. or “) 1/12 ft.
foot (ft. or ‘) 12in.
1/3 yd.
yard (yd.) 36 in.
3 ft.
rod (rd.), pole, or perch 161/2ft.
51/2 yd.
chain 4rd.
22 yd.
furlong 220 yd. or 40 rd.
or 10 chains or1/8 mi.
mile (mi.), statute 5,280 ft. or 1,760 yd.
or 8 furlongs
mile (mi.), nautical 2,025 yd.
Customary Unit Relation to Other of Measure Customary Units square inch (sq. in.) 0.007 (1/142) sq. ft.
square foot (sq. ft.) 144 sq. in.
square yard (sq. yd.) 1,296sq. in.
9 sq. ft.
square pole 301/4 sq. yd.
acre 43,560sq. ft.
40 rd. (1 furlong) x 4 rd. (1 chain)
square mile (sq. mi.) 640 acres
Linear Equivalents Area Equivalents
Fraction Decimal
Fraction to Decimal Equivalents
Customary Metric
Conversion Factors for Length
Metric Customary 1 micrometer (µm) 0.0000394 in. or 0.03937 mils 1 mm 0.0393701 in.
1 m 3.28084ft. or 1.09361 yd.
1 km 0.621371 mi.
Conversion Factors for Length
Metric Customary 1 mm2 0.001550 sq. in.
1 m2 10.7639 sq. ft. or 1.19599 sq. yd.
1 ha 2.47105 acres 1 km2 0.368102 sq. mi.
Conversion Factors for Area METRIC CONVERSION
Measurement and Geometr y 109
UNITS OF MEASURE: SI METRIC UNIT DATA
The General Conference on Weights and Measures (abbreviated to CGPM from the French Con-férence Générale des Poids et Mesures), which meets every four years concerning issues related to the use of the metric system, has established speciic rules for use, type style, and punctua-tion of metric units. The Napunctua-tional Institute of Standards and Technology (NIST)—formerly the National Bureau of Standards (NBS)—determines metric usage in the United States. Millimeters are the preferred unit for dimensioning buildings, with no symbol necessary if they are used consistently. Meters and kilometers are reserved for larger dimensions such as land surveying and transportation.
Unit names and symbols employ standard, lowercase type, except for symbols derived from proper names (for example, N for newton). Another exception is L for liter, to avoid confusing the lowercase l with the numeral 1. Preixes describing multiples and submultiples are also lowercase, except for M, G, and T (mega-, giga-, and tera-), which are capitalized in symbol form to avoid confu-sion with unit symbols, but which maintain the lowercase standard when spelled out. No space is left between the preix and the letter for the unit name (mm for millimeter and mL for milliliter). A space is left between the numeral and the unit name or symbol; for example, 300 mm.
Area Metric Equivalents
Although the United States and Canada mark the decimal with a point, other countries use a comma (for example: 5,00 vs. 5.00). For this reason, commas should not be used to separate groups of digits. Instead, the digits should be separated into groups of three, both to the left and the right of the decimal point, with a space between each group of three digits (for example, 1,000,000 is written as 1 000 000). This convention for metric units is used throughout the book.
Units with Compound Names
Area in Metrics
SI metric units for area, like volume, are derived from the base units for length. They are expressed as powers of the base unit: for example, square meter
= m2 = 106 mm2.
The square centimeter is not a recommended unit for construc-tion and should be converted to square millimeters.
The hectare is acceptable only in the measurement of land and water.
When area is expressed by linear dimensions, such as 50 x 100 mm, the width is written irst and the depth or height second.
Physical Quantity Unit Symbol
Area square meter m2
Volume cubic meter m3
Density kilogram per cubic meter kg/m3
Velocity meter per second m/s
Angular velocity radian per second rad/s Acceleration meter per second squared m/s2 Angular acceleration radian per second squared rad/s2 Volume rate of flow cubic meter per second m3/s Moment of inertia kilogram meter squared kg m2 Moment of force newton meter N m Intensity of heat flow watt per square meter W/m2 Thermal conductivity watt per meter kelvin W/m K Luminance candela per square meter cd/m2
SOFT AND HARD CONVERSIONS
Conversions of customary and SI units can be either “soft” or “hard.” In a soft conversion, 12 inches equals 305 millimeters (already rounded up from 304.8). In a hard conversion of this same number, 12 inches would equal 300 millimeters, which makes for a cleaner and more rational equivalency. This is the ultimate goal of total metrication within the building trades.
The process, however, is an extensive one, which will require many building products whose planning grid is in customary units to undergo a hard metric conversion of their own, making 6 inches equal 150 millimeters (instead of 152) and 24 inches equal 600 millimeters (instead of 610). Thus, to conform to a rational metric grid, the actual sizes of standard products such as drywall, bricks, and ceiling tiles will need to change.
Every attempt has been made in this book to represent accurately the relationship between customary and SI units. Except where noted, soft conversions are used throughout and, due to constraints of space, are usually written as follows: 1’-6” (457).
Measurement and Geometr y 111
METRIC CONVERSION
Inches and Millimeters Scale (1:1)
6” Feet and Meters Scale
➔
Feet Inches
Measurement and Geometr y 113
SLOPES AND PERCENTAGE GRADE
Note: Entries in blue indicate frequently used slopes. Slopes gentler than 1:20 do not require handrails; 1:12 is the maximum ADA-approved slope for ramps and 1: 8 is the maximum code-approved slope for ramps (non-ADA).
0.1 1:573.0 0.2 Degrees Gradient % Grade Degrees Gradient % Grade Degrees Gradient % Grade
90°
60°
45°
30°
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
1 2 3 4 5 6 7 8 9 10 12 15 20
Vertical Distance (V)
∞
Horizontal Distance (H)
Measurement and Geometr y 115
Calculating Slope Degrees:
Slope = Vertical Rise Distance (V)
= tangent m Horizontal Distance (H)
Slope Angle = tangent m Calculating Gradient:
= 1 unit in Horizontal Distance (H) Vertical Distance (V) Calculating % Grade: = 100 x Tangent (Slope) or 100 x V/H
Rectangle
perimeter = sum of length of sides
Quadrilateral
(Divide igure into two triangles and add their areas together.)
perimeter = sum of length of all sides c
perimeter = sum of length of all sides Trapezium
VOLUMES
Prism or Cylinder (right or oblique, regular or irregular)
volume = area of base x altitude Altitude (h) = distance between parallel bases, measured perpendicular to the bases. When bases are not parallel, then altitude = perpendicular distance from one base to the center of the other.
Pyramid or Cone (right or oblique, regular or irregular) volume = area of base x 1/3
Polygon Sides Area
Triangle (equilateral) 3 0.4330 a2
Square 4 1.0000 a2
Measurement and Geometr y 117
CIRCLE
Measurements 119
ELLIPSE
DOUBLE-CURVED SOLIDS Sphere
R
C
R
b Segment of Sphere
Sector of Sphere C
R
b b
c
a Ellipsoid
volume = 4 p R3 3 surface = 4 p R2
volume = p b2 (3R–b) 3 (sector – cone) surface = 2 p R b
volume = 2p R2 b 3 surface = p R(4b + C) 2 (segment + cone)
[BG-22-MS.live]
y
d
G
A B
E D x
F
p abc 6
surface = no simple equation volume =
perimeter (approximate) = p [1.5 (x + y) – !x y]
(Assume point G is the center point of the ellipse, with x,y coordinates of 0,0, and point B coordinates of Bx and By.)
area ABFEA =
(Bx x By) + ab sin-1 (Bx/a)
Measurement and Geometry 119