With old-fashioned "penny-farthing" bicycles, the crankarms were directly attached to the large drive wheel. One turn of the pedals moved the bicycle a distance equal to the circumference of the wheel. The larger the wheel, the farther the bicycle went for each turn of the pedals. The gear-inch system is a holdover from the days of the boneshakers, when wheel diameter determined the bike's "gearing" and was the key measurement on the bike.
Riding in a high gear on a modern bicycle is mechanically equivalent to riding a high-wheeler with a large wheel, where a low gear on a modern bicycle is the equivalent of a smaller wheel on a high-wheeler.
Gear inches express gear ratios in terms of the diameter of an equivalent directly-driven wheel, and are calculated as follows:
Diameter of drive wheel in inches × number of teeth in front chainring / number of teeth in rear cog.
gi = Gear Inches
dwd = Drive Wheel Diameter
fct = Front Chainring Teeth
rct = Rear Cog Teeth
For example, suppose the drive wheel is actually 26 inches in diameter. If the front chainring and rear cog have equal numbers of teeth, one turn of the pedals produces exactly one turn of the drive wheel, just as if the pedals were directly driving the drive wheel. That combination of gears and wheel is said to be "26 gear inches." If the front chainring has 48 teeth and the rear cog has 24 teeth, then each turn of the pedals produces two turns of the rear wheel. This is equivalent to doubling the size of the drive wheel; that is, it is like a directly-driven bicycle with a 52-inch wheel. That gear is said to be "52 gear inches."
A bicycle with a 26-inch wheel, a 48-tooth chainring, and a cassette with gears ranging from 11 to 34 teeth has a lowest gear of 26 × 48 / 34 = 36.7 gear inches and a highest gear of 26 × 48 / 11 = 113 gear inches.
For bicycles with 700c wheels, most cyclists quote gear inches based on a wheel diameter of 27 inches, corresponding to the old British tire size of 27 x 1¼" (ETRTO 630). This means that a 48/18 setup is usually considered to be 72". Strictly speaking, the rolling diameter of a 700c wheel is significantly lower, at about 26" for a 20mm tire or 26.3" for a 23mm tire. This means that the true gear on a 700c wheel can be as low as 69", which is the equivalent of only 46/18 on an actual 27" wheel, and can be the source of some confusion when comparing gears unless it is clear whether gear inches have been calculated using the actual wheel size or a conventionalized 27".
One could also calculate "gear centimeters," but in practice this is not done. An equivalent system customarily used by continental European cyclists is "meters of development", and measures the distance travelled with each turn of the crank. That is, meters of development is calculated as:
Circumference of drive wheel in meters × number of teeth in front chainring / number of teeth in rear cog.
Thus gear inches and development differ by a factor of π, and English/metric conversion. Some bicycles incorporate internally geared hubs, or other components that change the gear ratio and must be taken into account when calculating gearing.
Both "gear inches" and "meters of development" are concerned with the distance travelled per turn of the pedals, and are ultimately ways of indicating the mechanical advantage of the drivetrain, but neither of them take into account the length of the crankarm, which can vary from bike to bike. The crankarm is a lever arm. If two bicycles have different crank lengths but are otherwise identical, a longer lever arm gives a greater mechanical advantage. To take this into account, Sheldon Brown proposed a gear measurement system called "gain ratio," which is calculated by the distance travelled by the bike divided by the distance travelled by the pedals during one turn of the crank. He argued that it also has the advantage of being a pure number (with no units of measure); the calculation gives the same value whether it is carried out in inches or meters.