Topic 6 Measurement
Updated 2008-04-02 L
Please send comments to: J.Elfick@uq.edu.au
See also: Interesting websites
Table of contents
7.0 International system of units (SI), the 7 base units
2.0.1 Right-angled triangle
2.0.2 Golden mean, golden ratio
2.0.3 Greek alphabet, phonetic alphabet
2.0.5 Conic sections, parabola. ellipse, hyperbola
2.0.6 Parabola equation
2.0.7 Change scale of a map
2.0.8 Mathematics for science teachers
2.0.9 Handspans, trundle wheel
3.2.2 Rank scaling tables
2.1.6 Volume of liquid
3.2.2 Rank scaling tables
3.3.0 Accuracy and error
3.1.0 Counting, numerals, Roman numerals
3.2.0 Mass and weight
3.3.0 Length (l), the kilometre (km), metre (metre)
3.4.0 Area, square metre (m2), hectare
1.44 Area game (primary)
3.5.0 Volume (vol.), cubic metre (m3),
3.6.0 Estimating
3.7.0 Ratio and proportion, concentration, degrees proof
3.8.0 Angle, degree, arc minute, arc second, radian
3.9.0 CGS units (centimetre, gram, second)
3.10.0 The m.k.s. units
3.11.0 Imperial units used in land surveying (1 hectare = 10,000 m2, 1 kilometre = 1,000 m)
3.12.0 SI, CGS and FPS conversion, metric conversion
3.13.0 Energy conversion KJ, MJ, KWh, therm, BTU, calorie, horsepower
3.14.0 Oven temperatures
3.15.0 Measurement lessons for primary schools
19.8.0 Common measures
3.2.0 Mass and weight
Weight: 1000 grams (g) = 1 kilogram (kg), 1000 kg = 1 metric tonne (t)
3.2.1 Avoirdupois weight
3.2.2 Carat
3.2.3 Troy weight
3.2.4 Apothecaries' weight
3.2.5 Weights of one matchbox full of fertilizer
3.3.0 Length (l), the kilometre (km), metre (metre)
3.3.1 Angstrom unit
3.3.2 Astronomical unit, AU
3.3.3 United States lineal weights and measures
3.3.4 United States surface - land weights and measures
1.19 Length game (primary)
1.20 Pace distances (primary)
2.14 Measure in hand spans (primary)
4.17 Shapes game (primary)
3.5.0 Volume (vol.), cubic metre (m3),
3.5.1 Spoon volume
3.5.2 British liquid measures, imperial measures (fl. oz. = imperial fluid ounce)
3.5.3 American liquid measures, US measures, United States weights and measures, volume - liquid
3.5.4 Miscellaneous measures
3.5.5 Nautical measures
3.18 Volume of our fist (primary)
3.23 Volume of a liquid (primary)
3.7.0 Ratio and proportion, concentration, degrees proof
3.7.1 Concentration
3.7.2 Degrees proof, proof spirit
3.1.0 Counting, numerals
Roman numerals: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000
3.2.1 Avoirdupois weight, English and United States weights and measures
1 avoirdupois weight pound (lb) = 16 ounces (oz). All chemicals were sold by avoirdupois weight. (Latin: pondus - weight, 12 ounces of pure silver, 240 pennies, so cash to the value of 20 shillings sterling, symbol lb (Latin: libra - cash pound))
pound
 ounce
 drachm,
dram
 grain
(Troy)
 g
1
 16
 256
 7,000
 453.60
-
 1
 16
 437.5
 28.35
-
 -
 1
 27.34
 1.771845
A fluid dram is 1 ⁄ 8 of a fluid ounce, i.e. 3.696 mL USA and 3.551 mL UK. In Scotland, a dram is a small volume of Scotch whisky.
3.2.2 Carat
For precious stones,  1 carat is about 1 / 142 of an ounce. For gold,  a carat is a ratio of 1/24. Purity of gold is measured in carats. 24 carat gold is pure gold. 22 carat gold is 22 parts pure gold and 2 parts copper or other metal alloy.
The official mark stamped on gold and silver objects after being assayed is called the hall mark (from Goldsmith's Hall, London). For gold the standard mark is a crown in England for 22 and 18 carat gold followed by the number of carats in figures. Lower standards of gold have the number of carats in figures without the crown.
3.2.3 Troy weight
Gold is still sold in troy ounces, as were precious metals. 1 troy weight pound, lb = 12 troy ounces. 1 grain = 6.479 X 10-5 kg.
pound
 ounce
 pennyweight, dwt grain
 g
1
 12
 240
 5,760
 373.24
-
 1
 20
 480
 31.10
-
 -
 1
 24
 1.56
3.2.4 Apothecaries' weight, English and United States weights and measures
Apothecaries' measures were formerly used in pharmacy and were usually adopted in formulas. 1 fluid ounce = 8 drachms = 489 minums. The pound, ounce and grain are the same as in Troy weight.
In UK, the fluid drachm, fluidrachm = 3.55 mL.
pound
 ounce
 drachm
 scruple
 grain
 g
1
 12
 96
 288
 5,760
 373.24
-
 1
 8
 24
 480
 31,103
-
 -
 1
 3
 60
 3.888
-
 -
 -
 1
 20
 1.30
-
 -
 -
 -
 1
 0.06
3.2.5 Weights of one matchbox full of fertilizer
Ammonium sulfate (sulfate of ammonia) = 26 g
Potassium sulfate (sulfate of potash) = 40 g
Potassium chloride (muriate of potash) = 24 g
Single superphosphate, "super" = 22 g
Triple superphosphate, "super" = 20 g
Sulfur = 20 gm
3.3.0 Length (l), the kilometre (km), metre (metre)
Callipers, Vernier callipers, Vernier scale, callipers are for measuring internal and external diameters.
Gauge, feeler gauges, micrometer screw gauges
Find the thickness of one sheet of paper in a pile
Rule, measuring timber for carpentry, tape measure, dressmaking measurements - circumference of the chest / waist / hips, trundle wheels to measure the length of a crooked path
1 kilometre (1 kilometer) 1 km = 1000 metres (meters)
1 decimetre (1 decimeter) 1 dm = 0.1 metre (meter)
1 centimetre (1 centimeter) 1 cm = 0.01 metre (meter)
1 millimetre (1 millimeter) 1 mm = 0.001 metre (meter)
1 micrometre (1 micrometer) 1 mu m = 10-6 metre (meter)
1 nanometre (1 nanometer) 1 nm = 10-9 metre (meter) one billionth of a metre
1 picometre (1 picometer) 1 pm = 10-12 metre (meter)
3.3.1 Definition: 1 Angstrom unit, A = 10-10 metre, unit of measurement of wavelength, previous to SI unit
3.3.2 Definition: 1 Astronomical unit, AU = the mean distance between the Earth and the sun, about 149,598,000 km (92,956,000 miles). It is used as a convenient way to measure distance in the solar system.
3.3.3 United States lineal weights and measures
foot (singular), feet (plural) (mile - Latin: mille 1000, 1,000 paces, about 1680 yards)
mile
 furlong
 rod
 yard
 foot
 inch
1
 8
 320
 1,760
 5,280
 63,360
-
 1
 40
 220
 660
 7,920
-
 -
 1
 5.5
 16.5
 198
-
 -
 -
 1
 3
 36
-
 -
 -
1
 12
3.3.4 United States surface - land, weights and measures
1 square foot = 144 square inches
1 square yard = 9 square feet
1 square rod = 30.25 square yards
1 square rood = 40 square rods
1 acre = 4 square rods
1 square mile = 640 acres = 2,560 roods = 102,400 rods = 3,097,600 square yards = 27,878.400 square feet
acre
 rood
 rod
yard
 foot
1
 4
 160
 4,840
 43,560
-
 1
 40
 1,210
 10,890
-
 -
 1
 30.25
 272.25
-
 -
 -
 1
 9
3.4.0 Area, square metre (m2), hectare
Land: 100 metres (m) x 100 metres (m) = 10,000 square metres (m2) = (104 m2) = 1 hectare (ha) = 2.471 acre = 107,639 ft2
Imperial units used in land surveying (1 hectare = 10,000 m2, 1 kilometre = 1,000 m)
Area of cloth for a dress, area of a bolt of cloth, floor cover, area of a fitted carpet.
Irregular shape area - use of graph paper.
Regular shape area - square, rectangle, circle
Area of the top of a matchbox: 20 cm2
3.5.0 Volume (vol.), cubic metre (m3)
See diagram 2.1.6: Liquid volume
Volume in a measuring cylinder, meniscus
Volume of a bucket, fish tin, coconut, cups, a tablespoon, a teaspoon, of cooking oil to be used for food, of agricultural chemical to be used on a farm
Volume of water used at home or school, reading a water meter
Volume of petrol (gasoline) used by a motor vehicle
Volume of irregular shapes, volume of small quantity of sand or glass beads.
Displacement volume, overflow vessels
Volume of regular shapes, a cube, a block, cylinder, sphere, cone
Volume of gas used at home or school, reading a gas meter
Volume, solid: 1 centimetre (cm) x 1 centimetre (cm) x 1 centimetre (cm) = 1 cubic centimetre 1 cubic centimetre (cc, cm3) = 1 millilitre, 1 mL
Volume, liquid: 1000 millilitres = 1 litre (L)
3.5.1 Spoon volume
1 teaspoon (the smallest spoon) = 4.5 - 5 mL (0.2 fl oz) (1 fluid dram)
1 dessertspoon (the spoon you eat with) = 10 mL
1 tablespoon (tblspn) (spoon to serve with, the biggest spoon): 15 - 20 mL (0.5 fl oz)
1 teacup (cup to use with a saucer): 250 mL
1 matchbox volume: 25 mL
3.5.2 British liquid measures, imperial measures (fl. oz. = imperial fluid ounce)
These measures were usually adopted in formulas.
1 fluid ounce = 28.42 mL (0.96 US oz)
1 imperial pint = 568.3 mL (20 fl oz)
1 quart = 1140 mL (40 fl oz) (38.5 US oz)
1 imperial gill = 0.132 L (5 fl oz)
1 imperial gallon = 4.54609 litres
1 fluid drachm = 60 minims
1 fluid ounce = 8 fluid drachms
1 pint = 20 fluid ounces
1 gallon = 8 pints
3.5.3 American liquid measures, US measures, United States weights and measures, volume - liquid
1 liquid US pint = 473.1 mL ( 473.179 cc) (16 fl oz)
1 dry US pint = 550.6 mL (19 fl oz)
1 US fluid ounce = 29.56 mL (29.574 cc)
1 US gill = 0.118 L
1 US gallon = 3.79 L (3,785.435 cc)
1 pint = 4 gills
1 quart = 2 pints
1 gallon = 4 quarts (231 cubic inches)
3.5.4 Miscellaneous measures
1 wine glass - 1/4 cup
1 jigger = 1.5 fl oz
1 peck = 2 dry gallons
1 pinch or dash = what you pick up between your thumb and first two fingers
½ pinch = - what you can pick up between your thumb and one finger
1 salt spoon = 1/4 teaspoon
1 penny weight = 1/20 fl. oz
1 drachma = 1/8 oz
1 cupful = 284 mL
1 teacup (the cup you use with a saucer) = 200 mL
1 matchbox volume = 25 mL, Area of the top of a matchbox = 20 cm2
1 barrel (bbl) of crude oil = 42 US gallons, = 34.97 Imperial gallons (about 159.1 litres)
Foolscap printing paper = 13.5 X 17 inches
Foolscap writing paper = 13.25 X 16.5 inches
1 kati, caddy = 1 lb, 5 oz, 2 dr, weight still used in HongKong and Malaysia
1 cubit = 18 inches (English), 17.5 inches (Roman), 21 inches (Egyptian), traditionally measured from the tip of the elbow to the tip of the longest finger
1 hundredweight, 1 C-wt., 1 / 20 ton, long hundredweight (50.8 kg)
1 metric hundredweight 50 kg
1 US hundredweight 100 lb, short hundredweight (45.3 kg)
1 ell = 45 inches (English), 37 inches (Scotch), 54 inches (French)
1 hair breadth = 1 inch / 48
1 magnum = 2 English wine bottles (2 "reputed" quarts)
1 jeroboam = 4 English wine bottles = 4 X 262/3 fluid ounces
1 rehoboam =  6 English wine bottles
1 jerrican = 41/2 gallons (used for military fuel)
1 journey-weight of gold = 15 pounds troy (701 soverereigns)
1 nail = formerly a weight of 8 pounds or a length of 2.25 inches.
3.5.5 Nautical measures
Ship's cable was measured in shackles, 1 shackle = 12.5 fathoms
8 shackles, 100 fathoms, = 1 nautical mile / 10
The international nautical mile is 1852 m. The UK nautical mile is 1853.18 m (6,080 ft), its value in latitude 48o. A speed of one nautical mile per hour is called one knot.
3.6.0 Estimating
See 6.2:1 Estimating (primary)
Estimating of parameters, prediction, size perception, relative size
Estimating height of people, tree, a house, bridge, mountain
Estimating distance from the roadside, of the car ahead
3.7.1 Concentration
See also 7.7.0 Solutions, solubility, molar solution, solubility equilibrium, solubility product, solubility rules
See also 7.9: Amount of substance, the mole, molality, molarity, normality, standard solution
Concentration is the quantity of dissolved substance to quantity of solvent.
Dilution is the volume of solvent in which a measured amount of solute is dissolved.
Different ways of expressing concentration, e.g. ppm, normality, % weight for weight, % weight for volume
Different ways of expressing concentration (e.g. ppm, normality, % weight for weight, weight / weight, mass / mass %, weight for volume, w / v, weight / volume w / w. )
Parts per million, ppm, by mass, is about equivalent to a grain of sugar in a cup of tea, called milligrams per kilogram, or given as 0.0001%
Parts per million, ppm, 1 ppm = 1 mg per litre.
3.7.2 Degrees proof, proof spirit
Proof spirit contains, in Britain 49.28% alcohol (ethanol) by weight, 57.10% by volume, relative density 0.920 at 10.6oC (formerly specific gravity of 12/13 at 51oF), in USA 50% by volume at 15.6oC. This standard is quoted as 100 degrees of proof, 100o. If a spirituous liquor is p% overproof (above standard strength) it contains as much alcohol in 100 vol as in 100 + p vol of proof spirit. 20o proof = 0.2 X 57.1% alcohol = 11.42% ALC / VOL, e.g. white wine. Concentration of alcohol can also be measured with a hydrometer. Formerly proof spirit was that whic if poured over gunpowder and ignited would ignite the gunpowder. If the gunpowder didnot ignite, the spirit was under proof.
3.8.0 Angle, degree, arc minute, arc second, radian
Angle is the measurement of the inclination of one line to another. Measured in degrees, such that 360 degrees (360o) = 1 revolution. Also, measured in radians, such that 2 pi radians = 1 revolution. The degree can be divided into arc minutes, arcmin, such that 1' = 1/60 of a degree, and divided into arc seconds, arcsec, such that 1" = 1/3600 of a degree. Arc minutes and arc seconds are used in astronomy to measure the diameter or separation of astronomical objects.
3.9.0 CGS units (centimetre, gram, second)
Quantity CGS Unit Size
length centimetre 1 cm = 10-2 m
mass gram 1 g = 10-3 kg
area cm2 1 cm2 = 10-4 m2
volume cm3 1 cm3 = 10-6 m3
density g cm-3 1 g cm-3 = 10-3 kg m-3
3.10.0 The m.k.s. units
The metric system of units based on metre, kilogram, second. Also, the electrical unit was the ampere and magnetic constant was 4 pi X 10-7 Hm-1 (henry = H, now SI unit of inductance).
3.11.0 Imperial units used in land surveying (1 hectare = 10,000 m2, 1 kilometre = 1,000 m)
Imperial Metric Imperial Metric
1 square mile 258 9988 ha 1 link 0.201168 m (exact)
1 square mile 2.589988 km2 1 foot 0.3048 m (exact)
1 acre 4046.856 m2 1 mile 1.609344 m (exact)
2.471054 acres 1 ha 1 perch 25.2929 m2
1 rood 1011.714 m2 0.03954 perches 1 m2
1 ha = 2.471 acre = 107,639 ft2
1 yard = 0.9144 metre
1 acre = 0.404686 hectare
1 square foot = 0.92903 square metre
3.12.0 SI, CGS and FPS conversion, metric conversion
CGS = centimetre, gram, second
FPS = foot, pound, second
MKS or MKSA = metre, kilogram, second (ampere)
Physical quantity CGS unit FPS unit
length (m = metre) centimetre, 1 cm = 10-2 m foot, 1 ft = 0.3047075 m
" - inch, 1 in = 2.54 X 10-2 m
" - mile, 1 mile = 1.61 km
mass (kg = kilogram) gram, 1 g = 10-3 kg pound, 1 lb = 0.45359237 kg
" - ounce, 1 oz = 2.835 X 10-2 kg
" - ton, 1 ton = 1.016 X 103 kg
volume 1 cm3 = 10-6 m3 1 ft3 = 2.832 X 10-2 m2
" 1 litre, L =10-3 m3 1 in3 = 1.639 X 10-5 m2
" 1 millilitre, 1 mL = 1 cm3 -
density 1 g cm-3 = 10-3 kg m-3 -
velocity or speed 1 cm s-1 = 10-2 m s-1 -
" 100 km / hour 62.5 miles / hour
force dyne, 1 dyne = 10-5 N -
pressure, stress 1 dyne cm2 = 10-1 Pa -
" bar, 1 bar = 105 Pa 1 bar = 750.07 mm Hg
" millibar = 100 Pa -
energy, work (J = joule) erg, 1 erg = 10-7 J -
power (W = watt) 1 erg S-1 = 10-7 W horsepower, 1 hp = 745.7 W
viscosity poise, 1 P = 10-1 NM-2s -
thermal energy calorie, 1 cal = 4.17 J British thermal unit, 1 BTU = 1.055 X 103 J
3.13.0 Energy conversion KJ, MJ, KWh, therm, BTU, calorie, horsepower
1 KJ = 0.948 BTU
1 MJ = 948 BTU = 0.28 KWh = 0.37 horsepower hours
1 J = 0.239 calories
1 therm = 100,000 BTU = 106 MJ
1 KWh = 3412 BTU = 3.6 MJ
1 calorie = 4.187 J
1 horsepower = 746 watts
1 horsepower hour = 2.69 MJ
3.14.0 Oven temperatures
oC oF Gas mark Description
110 225 1/4 very cool, very slow
120 250 1/2 -
140 275 1 cool
150 300 2 -
170 325 3 very moderate
180 350 4 moderate
190 375 5 -
200 400 6 moderately hot
220 425 7 hot
230 450 8 -
240 475 9 very hot
3.15.0 Measurement lessons for primary schools
1.19 Length game (primary)
1.20 Pace distances (primary)
1.22 Compare different shapes (primary)
1.23 Make new shapes (primary)
1.44 Area game (primary)
2.14 Measure in hand spans (primary)
2.15 Measure with your body (primary)
2.16 Compare our weights (primary)
3.15 Record our heights (primary)
3.19 Single pan balance (primary)
3.21 Trundle wheel (primary)
3.22 Throw up and fall down (primary)
3.23 Volume of a liquid (primary)
4.14 Length game (primary)
4.15 Pace distances (primary)
4.17 Shapes game (primary)
4.18 Diameter of a thread (primary)
6.21 Estimating (primary)
6.22 Pendulum tells the time (primary)