Bicycle gear ratio, gear inches and development: three different numbers
Read front/rear tooth counts, distinguish dimensionless ratio from distance per crank turn, and keep wheel circumference and crank length in their own models.
Published 11 October 2026 · OpenBikeFit
One crank turn, two wheel turns
- Fictional front 42 / rear 21 = ratio 2.000.
- Rolling C = 2.160 m → development 4.320 m per crank turn.
- Equivalent-circle gear inches ≈ 54.1; not measured loaded diameter.
- At 80 rpm: conditional 20.736 km/h; not a speed observation.
Original schematic, not to scale. Crank length is separate. This model requires steady engaged rolling and no additional transmission stage.
On this page
- 01Start with the engaged drive, not the label
- 02Tooth ratio has no distance unit
- 03Development adds a rolling distance
- 04Gear inches need a named diameter
- 05Cadence turns development into conditional speed
- 06Crank length belongs to a separate convention
- 07Internal transmissions require another declared multiplier
- 08Carry a comparison that can be checked
Free gearing workspace
Compare complete gearing declarations
Retain every numerical tooth pairing, optional wheel quantities, source-specific constraints and unknowns. No ranking, chain-length instruction or compatibility verdict.
Open toolStart with the engaged drive, not the label
A gearing note needs the actual front chainring and rear sprocket in use, not just a cassette range or the number of available speeds. Name the bicycle configuration, exact part references and the tooth counts taken from their sources. A 12-speed label is a count of rear positions, not a numerical ratio. A chainring diameter measured with a ruler is not a tooth count either. Keep uncertain markings unresolved; a photograph of one exposed sprocket does not identify every concealed sprocket or the whole drivetrain.
This page uses an ordinary engaged chain drive with no additional hub or gearbox multiplier. During freewheeling, crank rotation and wheel rotation are not locked together by this model. Slipping, interrupted engagement and transient shifting are outside it. These assumptions matter before any equation is useful: a mathematically neat answer from the wrong transmission description is still the wrong reference.
Tooth ratio has no distance unit
Let F be front teeth and R rear teeth. The ideal engaged ratio q = F/R describes wheel revolutions per crank revolution. Fictional 42/21 gives q = 2.000: one complete crank turn advances the chain around the front ring and corresponds to two rear wheel turns in this model. Fictional 48/24 gives the same ratio. The two declarations are not the same parts and do not prove equal shifting, clearance or material condition.
Keep the original pair next to the quotient. An entry of 2.000 alone loses the ring and sprocket identities needed for later interface questions. Rounding a quotient for display is not permission to round a physical tooth count, which is an integer. Missing F or R leaves the ratio unknown, not zero, and a denominator of zero is not a usable gear.
Development adds a rolling distance
Declare rolling circumference C in metres for the stated wheel, tyre and observation condition. Development d = C × q is ideal distance travelled per crank revolution. With fictional C = 2.160 m and 42/21, d = 4.320 m. Keep the units alongside each number: entering 2160 as metres instead of millimetres creates a thousandfold error. Two equal tooth ratios on different rolling circumferences need not have equal development.
The existing wheel-circumference guide owns the measurement-reference distinction. Do not replace its rolling observation with bottom-bracket height, rim bead-seat diameter, tyre width or loaded axle height. If the circumference condition differs between configurations, disclose that difference instead of presenting a development difference as a pure gearing change. No tyre-pressure change or rolling test is prescribed by this article.
Gear inches need a named diameter
Gear inches are declared wheel diameter in inches multiplied by q. If a rolling circumference is deliberately converted to its equivalent circle diameter, D = C/π, then gear inches = C(mm)/(π × 25.4) × q. The fictional 2160 mm circumference and ratio 2.000 give approximately 54.1 gear inches. This is a circumference-equivalent diameter, not a measured loaded diameter or an automatic interpretation of a 700C, 29-inch or 650B label.
Use the same diameter convention when comparing two published gear-inch charts. A chart based on a nominal diameter and another based on observed rolling distance can differ even when the tooth counts agree. Preserve which convention each chart used. A discrepancy does not by itself establish a faulty component, inaccurate device or meaningful physical change.
Cadence turns development into conditional speed
For steady engaged rolling under the declared assumptions, speed in km/h = development in metres × cadence in crank revolutions per minute × 0.06. Fictional d = 4.320 m and cadence 80 rpm yield 20.736 km/h. This is a conditional calculation, not GPS, a speed-sensor observation, a target cadence or a prediction that a rider will sustain that speed on a hill.
Reversing the equation requires the same assumptions and a nonzero development. An observed speed while coasting cannot be used to infer active pedalling cadence from this relation. Record whether numbers came from a model, a device or an observation; identical digits do not make their origins interchangeable. Wind, slope, resistance, power and physiological response are not supplied by a tooth-ratio equation.
Crank length belongs to a separate convention
Crank length L is the radius of the pedal axis around the bottom bracket. It does not appear in q or d. Changing fictional L from 172.5 to 165 mm while keeping F, R and C fixed leaves both quantities unchanged. The pedal-axis path circumference, 2πL, changes instead. That distinction is why the existing crank-length tool compares geometry rather than silently changing a gear ratio.
Sheldon Brown proposed gain ratio as wheel radius divided by crank length, multiplied by the tooth ratio. With the same fictional circumference-equivalent radius 343.8 mm, the gain ratios are approximately 3.99 and 4.17 respectively. Label the radius convention and length units. Gain ratio is not a force measurement, joint-load result, efficiency score or reason to choose a crank for someone.
Internal transmissions require another declared multiplier
A hub or gearbox may add a gear-specific internal multiplier h. Then the ideal wheel/crank ratio is (F/R) × h, provided the manufacturer defines h in that direction. Some documents express the reciprocal or a different stage, so retain the exact convention and source. Do not assume direct drive for an unspecified hub or import a table from another product because the gear count matches.
Rohloff publishes technical data for its own hub, including separate internal gearing information and constraints. That is evidence for the named system, not a universal multiplier or permitted primary ratio for every hub. The ordinary derailleur example on this page is intentionally narrower. Leave additional stages unresolved until their actual documentation is available.
Carry a comparison that can be checked
A useful supplementary paper record keeps F/R, source identity, transmission type, circumference and its condition, any declared internal multiplier, and which derived quantity was requested. Retain raw numbers before rounding. Separate ratio, metres per turn, gear inches, rpm and km/h into named entries rather than putting them all in a field called gear size. Ask about unknowns before interpreting them.
Continue to the range-and-step guide to compare the whole numerical inventory, then identify the cassette interface and independent derailleur constraints. Reading this path does not approve replacement hardware or authorize installation. The blank worksheet contains general references and questions, not dedicated gear fields; use supplementary notes without relabelling a contact-point dimension. There is no account, saved-bike requirement or new measurement form in this guide.
Practical questions
Frequently asked questions
Is 42/21 the same numerical gear as 48/24?
Both tooth ratios equal 2.000 in the ordinary engaged model. Equal development additionally requires equal declared rolling circumference; parts, interfaces and shifting remain separate.
Does a shorter crank change gear inches?
Not if tooth counts and the declared wheel diameter are unchanged. Crank length enters a separately labelled gain-ratio convention, not ordinary gear inches or development.
Can I use 29 inches as the measured wheel diameter?
Not silently. That label is not a measurement of loaded diameter or rolling circumference. Record the diameter convention used by the chart or use a separately declared rolling reference.
Does the speed equation tell me my ideal cadence?
No. It is a steady engaged kinematic relationship under named assumptions, not a rider target, power model, terrain prediction or device measurement.