Gear Ratio and Gain Ratio Calculator
Gear Ratio and Gain Ratio Calculator
All four ways of describing one bicycle gear — gear ratio, gear inches, metres of development and Sheldon Brown’s gain ratio — with what each one ignores printed beside it. Only the gain ratio contains the crank length, which is why two bikes in the same gear can be geared differently.
Gain ratio, and the three measures that cannot see the crank
Wheel radius ÷ crank length × teethSheldon Brown’s own published road example: a 700 x 30 wheel of 680 mm outside diameter, 170 mm cranks, 53 teeth on the front and 19 on the rear
Formula
gear inches = wheel diameter in inches × gear ratio
development = π × wheel diameter in metres × gear ratio
gain ratio = (wheel radius ÷ crank length) × gear ratio
and the identity: gain ratio = development ÷ (2π × crank length)
- gain ratio
- Sheldon Brown’s. He gives it in two steps — “Find the radius ratio by dividing wheel radius by crank length”, which is done once per bicycle because it is the same in every gear, then multiply by front teeth over rear teeth — and says of the result that “This number is a pure ratio, the units cancel out”, so for every unit the pedal travels in its orbit the bicycle travels that many units
- the identity, and whose it is
- development ÷ (2π × crank length) gives the same number, because development is 2π × wheel radius × ratio and the 2π cancels. The page prints both and they agree to the last digit the engine carries. BROWN DOES NOT STATE THIS RELATION in either article read for this page, so it is presented here as arithmetic rather than as a quotation — and note that the divisor is the pedal circle’s circumference, 2π × crank length, not the crank length itself
- Brown on the other two
- of gear inches, “the handwriting is on the wall for all inch-based measurement systems”; of metres of development, “a bit more cumbersome” for needing a multiplication by π, “an irrational one, no less”. Development is nonetheless the only one of the four a governing body regulates
- what all four ignore
- drivetrain friction, which changes with how cross-chained the gear is, and tyre deflection under load, which makes the real rolling diameter smaller than any of these. All four are geometry rather than performance
Worked example
Sheldon Brown's own published road example: a 700 x 30 wheel of 680 mm outside diameter, 170 mm cranks, 53 teeth on the front and 19 on the rear
The wheel radius is 680 ÷ 2 = 340 mm, so the radius ratio is 340 ÷ 170 = 2.0 — the figure Brown prints
The gear ratio is 53 ÷ 19 = 2.7895, and 2.0 × 2.7895 = 5.58, which is the gain ratio Brown publishes for this gear
The same gear in the other three measures: 74.7 gear inches, a 2.1363 m wheel circumference and a development of 5.9591 m per crank revolution. The pedal circle is 2π × 0.170 = 1.0681 m round, and 5.9591 ÷ 1.0681 = 5.5789 — the identity closing on the headline exactly
Change the cranks to 175 mm and the gain ratio falls to 5.4195 while the other three do not move by a thousandth; 165 mm and it rises to 5.7480
Brown's second published example checks the other end of the range: a 26-inch wheel, 6¾-inch cranks and 46/16 gives 5.54, his printed figure
Set the sprocket, the wheel or the crank to zero and the panel shows nothing, because the division does not exist
One gear, four measures, four crank lengths
| Crank length | Gear ratio | Gear inches | Development | Gain ratio | Radius ratio |
|---|---|---|---|---|---|
| 165 mm | 2.7895 | 74.7 in | 5.9591 m | 5.7480 | 2.0606 |
| 170 mm | 2.7895 | 74.7 in | 5.9591 m | 5.5789 | 2.0000 |
| 172.5 mm | 2.7895 | 74.7 in | 5.9591 m | 5.4981 | 1.9710 |
| 175 mm | 2.7895 | 74.7 in | 5.9591 m | 5.4195 | 1.9429 |
One gear ratio, four wheels
| Wheel | Outside diameter | Gear ratio | Gear inches | Development | Gain ratio |
|---|---|---|---|---|---|
| 20 inch folder, 406 bead seat on 35 mm tyres | 476 mm | 2.7895 | 52.3 in | 4.1714 m | 3.9053 |
| 26 inch mountain bike, 559 on 50 mm | 659 mm | 2.7895 | 72.4 in | 5.7751 m | 5.4067 |
| 700C road, 622 on 28 mm | 678 mm | 2.7895 | 74.5 in | 5.9416 m | 5.5625 |
| 700C road, 622 on 32 mm | 686 mm | 2.7895 | 75.3 in | 6.0117 m | 5.6282 |
Why two bikes in 53/19 are not always in the same gear
This page is about cycling. Two different kinds of number appear in this category and they are not sourced the same way. A regulation — a minimum bicycle weight, a gear rollout limit for a junior rider — is published by a body for a period, and every one quoted here is printed with that body and that edition, because the Union Cycliste Internationale and the national federations revise them. A definition — a gear ratio, a circumference, a rate of climb — belongs to nobody and is the same everywhere, and where this page relies on one it says whose published wording it took it from.
Four measures are in circulation and they are not interchangeable. Gear ratio is chainring teeth divided by sprocket teeth, and it ignores the wheel, so it cannot compare a road bike with a 20-inch folder. Gear inches multiplies that ratio by the wheel diameter in inches and is an imperial relic: it is the diameter of the driving wheel a direct-drive high-wheel bicycle would need to be geared the same, which is why its numbers are in the seventies rather than anywhere meaningful. Development, or rollout, is the distance the bicycle travels in one crank revolution, wheel circumference times the ratio, in metres. Gain ratio is Sheldon Brown’s, and it is the only one of the four that is dimensionless and the only one that contains the crank length.
The reason to care about gain ratio is a fact the other three measures cannot express: two bicycles in the same gear with different crank lengths are not geared the same. Put a 53-tooth chainring and a 19-tooth sprocket on a 680 mm wheel and the gear ratio is 2.789, the gear inches 74.7 and the development 5.959 m — on 165 mm cranks, on 170 mm cranks and on 175 mm cranks alike, because none of those three numbers contains the crank at all. The gain ratio is 5.748, 5.579 and 5.419: the longer crank is the lower gear, by 5.7% from one end of that range to the other, and only one of the four figures shows it.
Where it comes from. Sheldon Brown set the measure out in two places read for this page — his own site and a two-page article for Adventure Cyclist — and the recipe is the same in both: divide the wheel radius by the crank length, which is a property of the bicycle and the same in every gear, then multiply by the chainring teeth over the sprocket teeth. Work in millimetres or in inches; it only matters that the wheel and the crank are in the same unit.
There is a second route to the same number, and this page prints it as a check rather than as a quotation. Development is 2π × wheel radius × ratio; the pedal travels 2π × crank length in one revolution; divide one by the other and the 2π cancels. So gain ratio = development ÷ the circumference of the pedal circle. Note what the divisor is: 2π × crank length, and not the crank length on its own — dividing by that alone gives a figure 6.283 times too large. Neither Brown article read for this page states this identity, so it is offered here as arithmetic, and the calculator prints both so you can see them agree.
One place the arithmetic stops being academic. A junior gear restriction is written as a limit on the development, in metres, and AusCycling publishes its own: 5.5 m for U9 to U13, 6.0 m for U15, 7.0 m for U17 and 7.93 m for J19, the last of those applying at all events from 1 January 2022, under its Technical Regulations: Road 1.02.03, which allows the rollout to be restricted “by the adjustment of the derailleur or other means”. That is a national federation’s rule for a stated period and not a law of cycling: a different federation publishes different figures, and this page prints AusCycling’s because they are the ones it read. The development row and the margin against 7.93 m are there so a parent with a commissaire’s roll-out board in front of them can see which gears are in and which are out.
All four are geometry and none of them says how hard the gear will feel, which depends on the rider and the gradient. For the speed a cadence in this gear produces, the cycling speed from cadence and gearing calculator carries the chain through to km/h; for the climb, the climbing gradient and VAM calculator does the rate of ascent.
Frequently asked questions
What is a gain ratio?
Sheldon Brown’s measure of a bicycle gear: the wheel radius divided by the crank length, multiplied by the chainring teeth divided by the sprocket teeth. It is dimensionless — it tells you how many units the bicycle moves for each unit the pedal travels in its orbit — so it is the only one of the four common measures that compares two bicycles fairly, because it is the only one that contains the crank length.
How do you work out a gear ratio?
Divide the chainring teeth by the rear sprocket teeth. 53 on the front and 19 on the back is 2.79, meaning the rear wheel turns 2.79 times for each turn of the cranks. That is all it tells you: it ignores the wheel size, so it cannot be compared between bicycles with different wheels, and it ignores the crank length.
Does crank length change the gearing?
Yes, and only the gain ratio shows it. In 53/19 on a 680 mm wheel the gear ratio is 2.7895, the gear inches 74.7 and the development 5.959 m whether the cranks are 165 mm or 175 mm, because none of those three contains the crank. The gain ratio is 5.748 on 165 mm cranks and 5.419 on 175 mm ones — a 5.7% difference, with the longer crank giving the lower gear.
What are gear inches, and why are the numbers so big?
Gear inches is the diameter in inches that the driving wheel of a direct-drive high-wheel bicycle would need to be geared the same way, which is why a normal road gear comes out around 75. It is a historical unit, it ignores crank length, and Sheldon Brown’s own view was that “the handwriting is on the wall for all inch-based measurement systems”.
What is rollout, and is there a limit on it?
Rollout, or development, is how far the bicycle travels in one crank revolution — the wheel circumference times the gear ratio, in metres. It is the quantity junior gear restrictions are written in: AusCycling publishes 5.5 m for U9 to U13, 6.0 m for U15, 7.0 m for U17 and 7.93 m for J19, the last applying at all events from 1 January 2022. Other federations publish their own figures, so check the rule for the event you are entering.
Related calculators
References
- Sheldon Brown. “Gain Ratios”, sheldonbrown.com/gain.html (read 15 Oct 2026; about 9,000 characters of which the article and its radius-ratio table are 5,000 to 6,000, and the table came through as readable TEXT). The source for gain ratio here. Two steps: “Find the radius ratio by dividing wheel radius by crank length”, done once per bicycle, then multiply by front teeth over rear. Of the answer: “This number is a pure ratio, the units cancel out”, so for each unit the pedal travels in its orbit the bicycle travels that many. BOTH published worked examples are reproduced through this site’s engine: a 700 x 30 road bike, 340 mm wheel radius, 170 mm cranks, radius ratio 2.0, 53/19 giving 5.58; and a mountain bike, 13 in radius, 6¾ in cranks, 46/16 giving 5.54. On the alternatives: “the handwriting is on the wall for all inch-based measurement systems”, and metres of development is “a bit more cumbersome” for needing a multiplication by π, “an irrational one, no less”.
- Sheldon Brown. “A new way to think about bicycle gears”, also headed “Mechanical Advantage Gain Ratios”, for Adventure Cyclist; the scan at sheldonbrown.com/adv-cycling/gainratios.pdf (read 15 Oct 2026; about 5,500 to 6,000 characters, pages numbered 30 and 31, NO DATE PRINTED in the text that came through, and no table came through as a table). Read as an independent statement of the same definition in a different publication: “Gain ratio is calculated by first dividing the wheel radius by the crank length”, the figure “expressed as a pure ratio”, the same 53/19 example giving 5.58, and gear inches dating from the high-wheel bicycle and so not accounting “for crank length”. IMPORTANT: NEITHER BROWN ARTICLE STATES the relation between gain ratio and metres of development. That identity — development divided by the pedal circle’s circumference, 2π times the crank length — is arithmetic, is labelled a derivation on the page rather than attributed to Brown, and is asserted exactly through the engine.
- Wikipedia. Gear inches (read 15 Oct 2026; about 10,000 characters). Cited for the DEFINITION — “the diameter in inches of the drive wheel of a penny-farthing bicycle with equivalent (direct-drive) gearing” — and for the exact factor 0.0254π between gear inches and metres of development. NOT cited for any formula: the article says its measures are “calculated as follows” and the equations are ABSENT from the fetched text, so only its prose was used. Its own references include the UCI’s Part 2 Road Races regulations of 1 January 2023 and the German and Austrian federations’ road regulations on junior gear restrictions, none of which was read here.
- AusCycling. “Updates: Junior gear restrictions – Road”, auscycling.org.au, article of 11 November 2021 (read 15 Oct 2026; about 10,000 characters, and the figures came through as readable text, as bold labels rather than a table). The one place in this batch’s reading where a governing body regulates the exact quantity the gearing page computes. “U9–U13: 5.5 metres”, “U15: 6.0 metres”, “U17: 7.0 metres”, “J19: 7.93 metres”, with “From 1 January 2022, the rollout of 7.93 metres will apply to J19 riders at all events”. It cites AusCycling Technical Regulations: Road 1.02.03 for restricting a rollout “by the adjustment of the derailleur or other means”. It does NOT say how a rollout is measured, and says nothing about the UCI’s own position, so no claim is made here about either.
- Wikipedia. ISO 5775 (read 15 Oct 2026; about 9,000 to 10,000 characters, designation table as readable TEXT). ISO 5775 was developed by the European Tyre and Rim Technical Organisation and is now managed by ISO; Part 1, “Bicycle tyres and rims: Part 1: Tyre designations and dimensions”, was last published August 2023 and Part 2, “Part 2: Rims”, March 2021. A clincher tyre carries its nominal section width and the nominal rim diameter in millimetres separated by a hyphen, so “32-597” is a 32 mm section on a 597 mm bead seat. Bead seat diameters: 622 mm (700C), 584 mm (650B, sold as 27.5 inch), 559 mm (26 inch mountain bike), 507 mm (24 inch), 406 mm (20 inch). The page warns that inch labels do not reliably indicate a bead seat diameter and that its own citations are to the superseded 1997 and 1996 editions, so a specific row should be checked against the standard.