What each Length unit is
Millimetre mm
Millimetres are what engineering drawings and datasheets use, because a whole number of them is precise enough for almost anything made by machine and avoids a decimal point that can be lost in a photocopy.
Centimetre cm
Centimetres are the everyday metric length — body measurements, paper, furniture — and they are the one metric unit that has no direct customary counterpart, which is why so much converting happens through them.
Metre m
The metre has been defined since 1983 as the distance light travels in 1/299,792,458 of a second. It is not a physical object any more; the platinum bar in Paris was retired.
Kilometre km
Kilometres are the road distance nearly everywhere outside the United States and the United Kingdom, which is why converting them is usually about reading a map or a race entry.
Inch in
An inch has been exactly 25.4 mm since 1959, when six countries agreed to stop using slightly different ones. Before that a US inch and a UK inch differed in the seventh significant figure: 25.400 05 mm against 25.400 00.
Foot ft
A foot is twelve inches, and it is the unit heights and room sizes are quoted in across the United States. Aviation quotes altitude in feet almost everywhere, including in countries that are metric for everything else; China is the exception, where controllers clear aircraft in metres.
Yard yd
A yard is three feet, and outside American football and fabric shops it is a unit most people rarely meet. It is defined as exactly 0.9144 metres.
Mile mi
A mile is 1,760 yards — a number that comes from a Roman thousand paces, adjusted by an Elizabethan statute to fit the furlong. It is exactly 1,609.344 metres.
Notes on the common Length conversions
cm to inches
American listings are written in fractions, not decimals
The decimal above is not the number on the label. American dimensions are written in binary fractions — halves, quarters, eighths, sixteenths — because that is how a tape measure is divided, so a sofa is 84 1/2 inches wide and a shelf is 11 3/4 deep. Converting 152 cm to 59.84 inches finishes the arithmetic and leaves you one step short of the mark you would look for on a tape.
That step is to put the part after the point over sixteen. Here 0.84 × 16 = 13.4, so 59.84 inches is 59 and 13/16, and since sixteenths are finer than anything you are buying gets built to, the sensible reading is 59 7/8. Rounding to the nearest eighth moves the figure by at most 1.6 mm. Rounding to the nearest quarter, which is how most furniture is described, costs 3 mm.
inches to cm
The inch became a metric unit in 1959
The inch has no independent existence any more. Since 1959 it has been defined as exactly 25.4 millimetres by agreement between the United States, the United Kingdom, Canada, Australia, New Zealand and South Africa — a decision to adopt a round metric value rather than to keep measuring the old ones against each other. Before that a US inch and a British inch differed in the seventh decimal place, which mattered only to people making gauge blocks and mattered a great deal to them.
One consequence runs through this whole page: a measurement in inches is already a metric measurement written in a different alphabet. Converting it loses nothing and adds nothing. Any imprecision in the centimetre figure came in with the original tape measure, not with the arithmetic.
km to miles
Three shortcuts for kilometres to miles, and what each one costs
Multiplying by 0.6 is the one everybody knows and the worst of the three. The true factor is 0.621371, so 0.6 is 3.4 per cent low: fine for 5 km, where it costs a tenth of a mile, and not fine for a 300 km drive, where it hides ten miles. The error is proportional, which means it grows exactly where you stop being able to check it.
Halving and adding an eighth is better and no harder. Half of a kilometre figure plus an eighth of it is 0.625 of it, which is 0.58 per cent high — an error of about six miles in a thousand. For 80 km that is 40 plus 10, giving 50 miles against a true 49.71. Over any distance a person drives in a day the shortcut and the calculator agree to the nearest mile.
miles to km
Where 1,609.344 comes from, one inch at a time
The mile is not measured against the metre; it is built from the inch. Since 1959 an inch has been exactly 25.4 millimetres, a foot exactly twelve of those, a yard exactly three feet, and a mile exactly 1,760 yards. Multiply the chain out and you get 1,609.344 metres with nothing trailing — a definition arrived at by stipulation, not a constant somebody refined with better instruments.
That is worth knowing because it settles a question this conversion invites. There is no more accurate value waiting to be discovered, and a converter showing 1.60934 or 1.6093 has rounded rather than disagreed. The digits stop after six decimal places in kilometres, which is a millimetre, and no measurement of any real mile has ever been that good.
cm to feet
The decimal above is one step short of a height
A height in feet is spoken as two numbers, and the field above gives one. Dividing 178 by 30.48 produces 5.84 feet, which is arithmetically correct and has never been said aloud by anybody describing a person. The form everybody uses is five foot ten, and reaching it means treating the whole number and the remainder as two separate things.
Keep the five and multiply the 0.84 by twelve: 10.1, so ten inches. The reason this cannot be skipped is that the remainder is a fraction of twelve, not of ten, so the digits after the point carry no information you can read directly. 0.5 means six inches, 0.25 means three, 0.75 means nine, and nothing else in the range maps onto anything you would guess.
cm to km
A map scale is this conversion, already done
A hundred thousand centimetres make a kilometre, and that is the number every metric map scale is built around. A 1:100,000 map puts one centimetre of paper on one kilometre of ground, so the ratio and the conversion are the same statement. Every other scale is a fraction of it: 1:50,000 gives two centimetres to the kilometre, 1:25,000 gives four, 1:10,000 gives ten.
Which means the arithmetic on a map is simpler than it looks. Measure the centimetres, divide by the number of centimetres in a kilometre at that scale, and you have kilometres. The only figure to carry is that one small number — two, four or ten — and it is printed on the sheet in the legend beside the ratio.