The formula, and where the odd-looking term comes from
A chain around two sprockets is two straight runs plus a partial wrap at each end. If both sprockets were the same size the answer would be simple: two centre distances of straight run, plus half the teeth of each sprocket, because between them the two wraps add up to one full sprocket. Written in pitches:
L = 2C ÷ p + (N1 + N2) ÷ 2
where C is the centre distance, p is the pitch and N1 and N2 are the tooth counts. The straight runs are not quite straight when the sprockets differ, though — the chain runs at a slight angle between a small sprocket and a large one, which makes the run longer than the centre distance. The correction for that is the third term:
L = 2C ÷ p + (N1 + N2) ÷ 2 + p × ((N2 − N1) ÷ 2π)² ÷ C
For a 15 and 45 tooth pair on a 5/8 inch chain at 23.5 inches of centre distance, that is 2 × 23.5 ÷ 0.625 = 75.2 pitches of straight run, plus 30 pitches of wrap, plus 0.625 × (30 ÷ 6.2832)² ÷ 23.5 = 0.606, giving 105.81 pitches. The nearest even length is 106.
Notice how small that third term is: six tenths of a pitch out of 106. It matters at the margin, when the exact figure lands near the halfway point between two even lengths, and it is otherwise almost invisible. On a machine where the two sprockets are close in size it disappears entirely.
Even numbers, and the size of a step
A roller chain alternates inner and outer plates, so a loop closed with a normal joining link has to contain an even number of pitches. Half links exist and they are a compromise nobody enjoys — a weaker plate in the run and an offset the sprocket does not love — so in practice chain comes in even counts and the length steps by two.
Two pitches of chain is one pitch of centre distance, because the chain goes out and comes back. On a 5/8 inch chain that is 15.9 mm of axle movement between one available length and the next. The adjuster slot has to cover at least that much, or there are centre distances that no chain length can reach with correct slack.
| Pitch | Chain sizes | Two pitches of chain | Axle movement |
|---|---|---|---|
| 1/2 in | 415, 420, 428 | 1.000 in | 12.7 mm |
| 5/8 in | 520, 525, 530, 532 | 1.250 in | 15.9 mm |
| 3/4 in | 630 | 1.500 in | 19.1 mm |
That column is the reason a new chain is fitted at the front of the adjustment range. It has to have somewhere to go as it wears, and the amount it has to give back before the next length becomes available is fixed by the pitch.
Adding teeth adds chain, predictably
Look again at the tooth term: the sum of the two sprockets, halved. Add two teeth to the rear and that term goes up by exactly one, so the chain needs one more pitch — half a link in the language people actually use, which is why two extra rear teeth so often means the same chain still fits, just further back in the slot, while four extra teeth means the next size up.
The front sprocket does the same thing in the same term, but a front sprocket change is a large change in ratio for a small change in length, which is one more argument in its favour when the adjustment range is tight.
Slack, and why the axle moves half of it
Free play at the midpoint of the lower run is measured as total movement, up and down. Push the chain up and the slack in the top run is taken out; let it down and it goes back. So the total free play is roughly twice the amount of extra chain length hanging between the sprockets, and pulling the axle back by one millimetre removes about two millimetres of measured slack.
Whether your machine wants that measurement taken on its wheels or on a stand, at which point along the run, and to what figure, is a per-machine specification. Some machines specify it with the swingarm at the position of maximum chain tension, which is nowhere near static ride height. This page does not know any of that and does not guess at it — it converts between a slack change and an axle movement, and the manual supplies everything else.
What can go wrong is abrupt
A drive chain is one of the few parts on a machine that can fail from neglect at full speed with no warning noise first. A chain that comes off can lock the rear wheel, punch through a crankcase, or wrap the sprocket. Pitch alone does not identify a chain — a 520 and a 530 share a pitch and differ in roller width and plate thickness, and the sprockets are cut for one or the other. Riveting a master link, setting slack, and checking sprocket wear are procedures with specifications, and both the specifications and the procedure belong to the manual and to whoever is qualified to carry them out. Everything on this page is arithmetic about length.
Questions people ask
How do I count the links on a motorcycle chain?
Count pitches, which is what the trade means by links on a motorcycle chain even though the word is used differently on a bicycle. Mark a roller with a paint pen or a zip tie, then count each roller as it goes past until you get back to the mark. A 120 link 525 chain has 120 rollers and 120 pitches, and since a pitch is five eighths of an inch it is 75 inches of chain end to end. Counting the outer plates instead gives you half the number, which is the usual source of confusion. The number is almost always even; if you count an odd number, the chain has an offset or half link in it somewhere, which is worth finding before you order a replacement.
How much does adding rear sprocket teeth change the chain length?
One pitch for every two teeth, straight out of the formula. The tooth-count term is the sum of both sprockets divided by two, so adding two teeth to the rear adds exactly one to it. Adding two teeth therefore needs one more pitch of chain, which is not a length you can buy, so the axle takes it up instead — about eight millimetres of adjuster on a 5/8 inch chain. Adding four teeth needs two more pitches, which is the next available length. There is also a second, smaller effect: making the sprockets more different in size increases the correction term, but on a typical machine that adds well under a tenth of a pitch and disappears in the rounding.
Can I use a chain with an odd number of links?
It requires an offset link, sometimes called a half link, and it is a compromise rather than a normal fitment. The offset link uses a cranked plate to join an inner end to an inner end, and that cranked plate is the weakest part of the loop and the part most affected by the load the drive puts through it. Manufacturers of chain generally state that they are not intended for high-load applications, and many machines are never supplied with one from new. When the arithmetic lands between two even lengths, the usual answer is to take the shorter even length and let the axle sit further forward, or the longer one and let it sit further back, rather than to split the difference with an offset link.
Why does the axle end up so far back with a new chain?
It should not — a new chain of the right length normally sits near the front of the adjustment range, because the whole point of the range is to have somewhere to go as the chain wears. If a new chain puts the axle at the back, the usual causes are a chain that is two pitches short for the sprockets fitted, sprockets with more teeth than the ones the chain was sized around, or a chain that is not actually new. Run the tooth counts and your measured centre distance through the arithmetic and compare the exact figure against the length you bought. Chain wear happens at the pins and bushings, which is why a worn chain measures longer than a new one over the same number of pitches and why the axle keeps walking back over the life of the chain.
Do I measure centre distance with the machine on a stand?
For this calculation, measure it at the axle position you intend to end up at, with the chain slack rather than under tension, because what the formula wants is the geometric distance between the two shaft centres. What complicates it on a real machine is that the centre distance changes as the suspension moves — the swingarm pivots around a point that is not the gearbox output shaft, so the distance grows and shrinks through the travel, and that is exactly why chain slack exists and why some manuals specify measuring slack at a particular suspension position rather than at rest. If you are sizing a chain, the resting figure is the one to use here. If you are setting slack, the manual position governs.