THE GEARING MATHS
Teeth, turns
and distance.
Gearroll calculates the kinematic relationship for one engaged bicycle chainring and rear sprocket, using your entered wheel rolling circumference and crank cadence.
Four entries with different jobs
The front and rear tooth counts establish how many wheel revolutions correspond to one crank revolution. Circumference turns those wheel revolutions into ground distance. Cadence supplies the number of complete crank revolutions per minute.
Garmin’s cadence explanation defines cycling cadence using crank revolutions per minute. It is not a count of separate left and right downstrokes.
Garmin’s wheel circumference guidance describes manual circumference entry and measuring the wheel as an option. Gearroll deliberately requests your circumference rather than choosing a value from a nominal tyre label. The prefilled 2100 mm is an example only.
Why front teeth ÷ rear teeth?
One complete front chainring revolution moves one chain pitch for every front tooth. Dividing that movement by the rear sprocket’s tooth count gives its revolutions. With a directly engaged rear wheel, that is also the wheel’s number of revolutions. This is a simple chain-motion derivation, not a drivetrain efficiency model. SRAM’s manufacturer discussion of tooth counts and gear ratios gives gearing context; no performance or product-selection claim from it is used here.
Gear ratio = front teeth ÷ rear teeth
Rollout (m/crank revolution) = circumference (m) × gear ratio
Wheel rpm = crank cadence × gear ratio
Speed (km/h) = rollout × crank cadence × 60 ÷ 1000
Speed (mph) = rollout × crank cadence × 60 ÷ 1609.344
One international mile is 1609.344 m; one inch is exactly 0.0254 m. Unit changes retain the canonical circumference internally. Editing the converted number establishes a new circumference, rather than silently retaining the previous measurement.
Worked example: 48 front, 16 rear
| Illustrative circumference | 2100 mm = 2.1 m |
|---|---|
| Ratio | 48 ÷ 16 = 3 wheel turns per crank turn |
| Rollout | 2.1 × 3 = 6.3 m per crank turn |
| At an entered 90 rpm | 6.3 × 90 = 567 m/min |
| Theoretical speed | 567 × 60 ÷ 1000 = 34.02 km/h, approximately 21.13905 mph |
Change the rear sprocket, keep the inputs visible
With the same 48-tooth front ring, 2.1 m circumference and 90 rpm cadence, a 24-tooth rear sprocket gives ratio 2, rollout 4.2 m and theoretical speed 22.68 km/h. Those are arithmetic comparisons. This tool does not check whether a 24-tooth sprocket, chain length or any replacement component fits your bicycle.
A repeating ratio stays a ratio
For front 50, rear 17 and circumference 2.1 m, the ratio is exactly 50/17 and rollout is exactly 105/17 m per crank turn, approximately 6.176470588 m. At 90 rpm the speed is exactly 567/17 km/h, approximately 33.352941176 km/h. Gearroll multiplies the exact fraction; it does not round 50/17 first.
Driven speed is not coasting speed
A freewheel can allow a bicycle to roll while the cranks are stationary. Zero entered cadence produces zero driven wheel rate and zero driven speed in this model; it does not claim a coasting bicycle must stop. Tyre slip, internal hub gears, motor reductions, differential drives and other mechanisms are not modelled.
Wind, gradient, surface, rider power, resistance and losses are not inputs. There is no attainable-speed forecast, route timing, effort estimate, safety judgement, cadence recommendation or drivetrain compatibility advice. Circumference uncertainty passes directly into rollout and theoretical speed.
Exports and software limits
Copy, Text and CSV are local actions. Exports include exact fractions plus decimal approximations rounded to at most 12 places, with canonical circumference in metres and clear model limits. A fraction such as 105/17 has no finite decimal expansion; the decimal is deliberately marked as an approximation.
Supported input bounds are 0.001–100 m circumference, whole tooth counts from 1 to 1000, and 0–1000 rpm cadence. These are software bounds, not suggested bicycle designs or riding ranges. Number text is limited to 100 characters, with decimal exponents between −30 and +30. Tooth counts require whole digits, without commas, decimal notation or exponents.