bikegearing

Gain ratio calculator

Work out gain ratio from your chainring, cog, tyre and crank length — the one gearing number that accounts for the lever you actually push.

155 to 180 mm
20 to 70 teeth
9 to 52 teeth

—gain ratio

Ratio
—
Wheel radius
— mm

Source for 700 × 28c (28-622): ETRTO 28-622 — the designation on the tyre sidewall. The wheel radius is derived from it, not tabulated — half the bead-seat diameter plus the nominal section width — and the crank length is measured centre to centre, from the bottom bracket axle to the pedal axle.

339.0 mm
your wheel radius, the outer circle
172.5 mm
your crank length, the dashed circle a pedal sweeps
Your two lengths on one scale: the outer circle is the wheel's radius, the dashed inner circle the path a pedal sweeps — one crank length. The circles are not the gain ratio. What they show is the ratio of the two radii, 339.0 mm against 172.5 mm, or 1.97 — one of the two factors a gain ratio is built from. The other is the drivetrain ratio, 4.55, and no circle here can show it: the 8.93 above is those two multiplied. Each figure is rounded for display, so multiplying the printed ones back lands a digit out. The outer circle is drawn the same size whatever wheel you choose, because the only thing these two lengths jointly say is their ratio — it is the inner circle that moves. And they are drawn about one centre so the lengths can be compared side by side: on a bicycle the wheel turns about its hub and the cranks about the bottom bracket, which are nowhere near each other.

What gain ratio means

Gain ratio is how far the bike travels for every unit of distance your foot travels around the pedal circle. Turn the cranks through one full circle and your foot covers the circumference of that circle; the bike covers rather more ground than that. Gain ratio is the second divided by the first.

It is the wheel’s radius measured against the crank’s length — the two levers, one against the other — multiplied by the ratio of chainring teeth to cog teeth:

R = r l × n c n s

where R is the gain ratio, r the wheel radius, l the crank length, nc the chainring teeth and ns the rear cog teeth.

Gain ratio has no unit, and that is a fact about the arithmetic rather than a convention. Both lengths are in millimetres, so the unit divides out of the first fraction; tooth counts have no unit to begin with. Nothing on this page is converted to inches, and nothing needs to be. A 50 × 11 on 700 × 28c with 172.5 mm cranks comes out near 8.9; the same bike in 34 × 34 is near 2.0.

Why it beats gear inches

Gear inches ignores crank length. That is the one thing it genuinely gets wrong, and it is not a rounding matter: a crank is a lever, and two riders on identical gear inches with different cranks are not pushing the same gear. The rider on shorter cranks has less leverage, so the same gear is harder to turn — which is exactly what a longer crank buys on a climb, and exactly what gear inches cannot see.

So gain ratio is the number to compare when the bikes differ. Two bikes with the same gear inches and different cranks have different gain ratios, and the difference is real; two bikes with the same gain ratio feel the same at the pedal even if their wheels, rings, cogs and cranks are all different. That is the whole reason the quantity is worth a page: it is the comparable one.

The trade is legibility. Gear inches is a diameter you can picture — the wheel of the penny-farthing that would give you the same gear. A gain ratio of 8.9 is not a picture of anything, which is why gear inches has outlived the machinery it described.

Where the wheel radius comes from

The radius is derived from the two ETRTO figures printed on your tyre’s sidewall, not tabulated: a bead-seat diameter b and a nominal section width s give a diameter of b+2⁢s, and the radius is half of it.

r = b + 2 ⁢ s 2

This assumes the tyre’s section height equals its nominal width — an aspect ratio of about 1, which is an approximation rather than a measurement, and it is the same assumption behind every figure on this site. The gear inches page states where it differs from published rollout tables and why; the same caveat applies here, and for the same reason.

Crank length is measured centre to centre: from the bottom bracket axle to the pedal axle. It is usually stamped on the inside of the crank arm, in millimetres. If yours is not, measure it rather than guessing: it divides straight into the answer, so an error in it is an error of the same proportion in the gain ratio.