Bicycle Gear Maths: Ratios, Cadence & Speed

Mathematics becomes wonderfully tangible on a bicycle. A rider turns the cranks, a chain carries force to a rear sprocket, the wheel rotates, and the bicycle moves. Beneath that familiar action are ratios, circumference, angular speed, torque, power, percentage gradient and careful measurement. This is why mathematics is important: it lets us connect what our legs feel to what the bicycle actually does.

For students searching for maths in everyday life, bicycle gearing is a particularly honest example. A smaller gear may make each pedal turn easier but cover less ground. A larger gear may cover more ground but demand more torque. Neither is automatically “better”. The sensible choice depends on slope, speed, wind, surface, bicycle, rider and purpose.

This article develops the mathematics from first principles. The numbers are illustrative rather than prescriptions for training or equipment. Real tyres deform, drivetrains lose energy, riders vary, traffic changes and safe riding always matters more than completing a calculation.

Why bicycle gearing is mathematical

A bicycle is a system of linked rotations. The front chainring rotates with the cranks. Its teeth pull the chain. The moving chain turns a rear sprocket, which usually turns the rear wheel. Counting teeth turns this mechanism into a ratio problem.

Suppose a chainring has 48 teeth and a rear sprocket has 16 teeth. One complete crank revolution advances 48 chain links past the chainring. Because the rear sprocket needs 16 links for one revolution, it turns 48 ÷ 16 = 3 revolutions. In an ideal rigid system, the rear wheel therefore turns three times per crank revolution.

That simple division is the bridge between algebra and motion. It also shows an important habit in mathematics education: identify what one complete cycle does before trying to calculate a whole journey.

Four quantities that must not be confused

Quantity	Meaning	Typical unit
Gear ratio	Front teeth divided by rear teeth	No unit
Cadence	Crank revolutions per minute	rpm
Wheel circumference	Distance around the rolling wheel	metres
Road speed	Distance travelled per time	m/s or km/h

The quantities interact, but they are not interchangeable. Cadence is not speed. A gear ratio is not a distance. Wheel diameter is not wheel circumference. Clear units stop many errors before they begin.

Did you know? One ratio can describe two opposite experiences

A large front-to-rear tooth ratio gives more wheel rotations for each crank turn, so the bicycle travels farther per pedal revolution. The corresponding torque multiplication from crank to rear hub is smaller in the idealised model. A low ratio gives fewer wheel rotations but greater torque multiplication. Speed advantage and force advantage trade places.

This is a recurring mathematical idea. Levers, pulley systems, electrical transformers and financial trade-offs all reward the same question: what is gained, and what is given up?

The chain-drive ratio

Let F be the number of teeth on the front chainring and R the number on the selected rear sprocket. The ideal wheel revolutions per crank revolution are

gear ratio = F ÷ R.

For 50 teeth at the front and 20 at the rear, the ratio is 2.5. For 34 and 28, it is about 1.214. British Cycling’s explanation of gears uses the same tooth-count logic and notes, for example, that a 50:12 combination turns the wheel a little more than four times per crank revolution. Its practical guidance also distinguishes gear choice from cadence rather than treating them as the same variable.

Ratio notation needs interpretation

Cyclists may write 50:20, say “two point five to one”, or simply call it a 2.5 ratio. The colon records two counts; the quotient describes the rotational relationship. Reducing 50:20 to 5:2 preserves the ratio but hides the actual components. Both representations are useful for different questions.

If the task is to compare motion, 5:2 is enough. If the task is to identify a physical sprocket, the original tooth counts matter. Mathematics compresses information, but a modeller must notice what was compressed away.

Comparing adjacent gears

Imagine a 42-tooth chainring with rear sprockets of 14, 16, 18 and 21 teeth.

Rear teeth	Ratio	Change from previous ratio
14	3.000	—
16	2.625	−12.5%
18	2.333	−11.1%
21	2.000	−14.3%

The tooth counts rise by 2, 2 and 3, yet the percentage changes in ratio are not equal. That is why a cassette’s spacing influences how smoothly a rider can adjust cadence. Equal additive steps do not create equal proportional steps.

To calculate a percentage change from an old value to a new value, use

percentage change = (new − old) ÷ old × 100%.

The sign tells the direction. The magnitude tells the proportional size.

Gear inches and metres of development

Two traditional summaries include wheel size as well as tooth ratio. Gear inches multiply the nominal wheel diameter in inches by F ÷ R. Metres of development multiply the rolling circumference in metres by F ÷ R.

Metres of development are especially direct: they estimate how far the bicycle travels for one crank revolution. If measured circumference is 2.10 m and the ratio is 2.5, development is 5.25 m per crank revolution.

These measures help compare bicycles with different wheel sizes. A ratio alone cannot do that. A folding bicycle and a road bicycle can share the same tooth ratio yet travel different distances per crank turn because their wheels have different circumferences.

From cadence to road speed

Cadence counts how many crank revolutions occur each minute. If development is known, the ideal distance per minute is

distance per minute = cadence × development.

To convert metres per minute to metres per second, divide by 60. To convert metres per second to kilometres per hour, multiply by 3.6. Combining the steps gives

speed in km/h = cadence × circumference × gear ratio × 60 ÷ 1000.

Here circumference is in metres and cadence is in revolutions per minute.

A dimensional check

The units reveal whether the formula makes sense:

rev/min × m/wheel rev × wheel rev/crank rev.

The revolution labels cancel, leaving metres per minute. Multiplying by 60 min/h and dividing by 1000 m/km leaves km/h. Unit cancellation is not decorative; it is a compact proof that the conversion structure is coherent.

Example: a calm-road estimate

Take a 46-tooth chainring, an 18-tooth sprocket, a measured wheel circumference of 2.08 m and cadence of 80 rpm.

Gear ratio = 46 ÷ 18 = 2.556.
Development = 2.08 × 2.556 = 5.316 m per crank revolution.
Distance per minute = 80 × 5.316 = 425.28 m.
Speed = 425.28 × 60 ÷ 1000 = 25.52 km/h.

This is an ideal kinematic estimate. It says what rolling speed corresponds to the measured cadence and assumed circumference if the tyre rolls without meaningful slip. It does not say how much effort is required to hold that speed.

Solving backwards

Algebra makes the relationship useful in both directions. If a rider wants to know the cadence corresponding to a measured speed,

cadence = speed in km/h × 1000 ÷ (60 × circumference × gear ratio).

At 24 km/h, circumference 2.08 m and ratio 2.556, cadence is about 75.2 rpm. Rearranging a formula is more than an examination manoeuvre: it changes which measurement can be inferred from the others.

Why GPS speed and calculated speed may disagree

The wheel’s loaded rolling circumference may differ from a label. Tyre pressure, load and surface can change it slightly. A cadence sensor may sample or smooth data. GPS speed can fluctuate because of positioning and filtering. Coasting breaks the cadence–speed link on a freewheel bicycle because the wheel continues while the cranks stop.

Disagreement is therefore an invitation to inspect assumptions, not immediate proof that one device is defective.

Torque and mechanical advantage

Speed tells only half the story. Torque describes the turning effect of a force. In a simple model,

torque = perpendicular force × lever arm.

If a rider applies a 180 N tangential force to a 0.17 m crank, crank torque is 30.6 N·m. That torque creates chain tension at the front chainring. Ignoring losses, a smaller rear sprocket produces lower rear-hub torque than a larger rear sprocket for the same crank torque.

An ideal relationship is

rear-hub torque = crank torque × R ÷ F.

With 34 front teeth, 28 rear teeth and 30.6 N·m at the crank, rear-hub torque is 30.6 × 28 ÷ 34 = 25.2 N·m. With 50 front and 12 rear, it would be only 7.34 N·m. The higher gear turns the wheel farther but multiplies torque less.

Force at the tyre

If wheel radius is r, the ideal tangential driving force at the tyre contact is approximately

driving force = rear-hub torque ÷ r.

For 25.2 N·m and a 0.335 m wheel radius, the force is about 75.2 N. This is not necessarily the bicycle’s net forward force. Rolling resistance, aerodynamic drag, slope and acceleration also matter.

Power links torque and cadence

Rotational power can be expressed as

power = torque × angular speed.

Cadence must be converted from revolutions per minute to radians per second:

angular speed = cadence × 2π ÷ 60.

At 80 rpm, angular speed is about 8.38 rad/s. If average effective crank torque were 25 N·m through the simplified cycle, power would be about 209 W. Real pedal torque varies through each revolution, so instrumented power meters estimate the time-varying relationship more carefully.

Why a low gear can feel easier without reducing the hill’s energy requirement

Raising a bicycle and rider through a vertical height h changes gravitational potential energy by approximately mgh. A lower gear does not erase that energy. Instead, it lets the rider deliver the work through more pedal revolutions, generally with less force per revolution and more time.

This distinction between force, work and power is fundamental:

Force concerns push or pull.
Work concerns force acting through distance.
Power concerns the rate at which work is done.

A gear changes the relationship between force and distance. It does not create energy.

Climbing and gradient

Road gradient is commonly reported as a percentage:

gradient = vertical rise ÷ horizontal run × 100%.

A road rising 30 m over a horizontal 600 m has a 5% gradient. This is not the same as a 5° angle. The angle θ satisfies tan θ = rise ÷ run, so θ = arctan(0.05), about 2.86°.

Estimating climbing power

At constant speed on a simplified slope, the power needed to gain height is

gravitational power = mass × g × vertical speed.

If road speed is v and slope angle is θ, vertical speed is v sin θ. For modest gradients, sin θ is close to the decimal gradient, though not exactly identical.

Consider a combined bicycle-and-rider mass of 75 kg travelling at 12 km/h, or 3.333 m/s, on a 5% gradient. Using the small-slope approximation, vertical speed is about 0.1667 m/s. Gravitational power is about 75 × 9.81 × 0.1667 = 123 W. Rolling resistance, air drag and drivetrain loss would add to the required rider power.

Choosing a gear for a target cadence

Suppose a student models a climb at 12 km/h with wheel circumference 2.10 m and wishes to compare gears at 80 rpm. Required development is

12,000 m/h ÷ 60 ÷ 80 = 2.5 m per crank revolution.

The required gear ratio is 2.5 ÷ 2.10 = 1.190. A 34-tooth chainring with a 29-tooth rear sprocket gives 1.172; with 28 teeth it gives 1.214. Neither is exactly 1.190, because physical gears are discrete. The rider chooses the nearby option that suits conditions.

This is a small optimisation problem with constraints. Mathematics narrows the choices but does not make the personal decision.

Worked examples

Worked example 1: compare two ratios

Gear A is 48:16 and Gear B is 36:24.

Gear A ratio = 3.0.
Gear B ratio = 1.5.
Gear A turns the wheel twice as many revolutions per crank turn as Gear B.
In the ideal torque relation, Gear B gives twice the rear-hub torque for the same crank torque.

“Twice the ratio” must be attached to a specific outcome. It does not mean twice the real speed on any road because cadence and resistance may change.

Worked example 2: calculate development

Circumference is 2.06 m, front ring 40 teeth and rear sprocket 20 teeth.

Ratio = 40 ÷ 20 = 2.
Development = 2.06 × 2 = 4.12 m.

Each ideal crank revolution advances the bicycle about 4.12 m.

Worked example 3: predict speed

Use the previous gear at 75 rpm.

Distance per minute = 4.12 × 75 = 309 m.
Speed = 309 × 60 ÷ 1000 = 18.54 km/h.

Report a sensible number of significant figures. A prediction of 18.540000 km/h would imply unjustified precision.

Worked example 4: find cadence from speed

At 27 km/h, development is 5.8 m.

Metres per minute = 27,000 ÷ 60 = 450.
Cadence = 450 ÷ 5.8 = 77.6 rpm.

The estimate assumes engaged pedalling rather than coasting.

Worked example 5: compare a gear step

Changing from a 17-tooth to a 19-tooth rear sprocket while keeping a 46-tooth front ring changes ratio from 2.706 to 2.421.

Proportional change = (2.421 − 2.706) ÷ 2.706.
Percentage change ≈ −10.5%.

At the same road speed, cadence would need to rise by the reciprocal effect, not simply by 10.5%. Exact cadence ratio is 2.706 ÷ 2.421 ≈ 1.118, an 11.8% increase.

Worked example 6: calculate crank torque

A tangential pedal force of 150 N acts at a crank length of 0.175 m.

Torque = 150 × 0.175 = 26.25 N·m.

If the force is not perpendicular, use the perpendicular component. A force aligned with the crank produces no turning moment about the crank axis.

Worked example 7: calculate ideal wheel force

Use 26.25 N·m crank torque, 32 front teeth, 28 rear teeth and wheel radius 0.34 m.

Hub torque = 26.25 × 28 ÷ 32 = 22.97 N·m.
Tyre force = 22.97 ÷ 0.34 = 67.6 N.

This is the modelled driving force before subtracting resistive forces.

Worked example 8: energy gained on a hill

A combined mass of 70 kg gains 45 m of height.

Potential-energy increase = 70 × 9.81 × 45.
Result ≈ 30,902 J, or 30.9 kJ.

This mechanical value does not equal food energy used by the body; human efficiency and other losses must be considered.

Worked example 9: detect a circumference error

A calculation uses 2.20 m circumference, but a loaded rollout measures 2.10 m.

Relative error = (2.20 − 2.10) ÷ 2.10 × 100%.
Error ≈ 4.76% high.

Because predicted speed is directly proportional to circumference, the speed prediction is also about 4.76% high if all else is unchanged.

Worked example 10: choose between two available sprockets

Required ratio is 1.25 with a 35-tooth chainring. Candidate sprockets are 28 and 30 teeth.

35 ÷ 28 = 1.25 exactly.
35 ÷ 30 = 1.167.

The 28-tooth option matches the target ratio, but a real choice could still favour another gear because of cadence preference, terrain change or chainline. Solving one equation does not represent the entire ride.

Measurement, modelling and limits

Good mathematics makes assumptions visible. The chain-ratio equations are kinematic and idealised. They describe linked rotations well, but real motion includes additional mechanisms.

Measuring rolling circumference

A classroom investigation can mark the tyre and floor, load the bicycle normally, roll one or several wheel revolutions in a straight line, and measure distance. Dividing a multi-revolution distance by the number of revolutions can reduce the influence of a small mark-position error.

Repeat the measurement. Report the mean and range. State tyre pressure, load and surface. Those details make the result reproducible and teach more than copying a nominal tyre label.

Drivetrain efficiency

Chains, bearings, tyre deformation and misalignment dissipate some energy. Efficiency can be represented as useful output power divided by input power. It is usually less than 1. A fixed percentage is only a model; losses can change with maintenance, load, chain angle and component condition.

Aerodynamic drag

At higher speed, air resistance becomes important. A simplified drag model is proportional to one half times air density, drag coefficient, frontal area and speed squared. The power used against that drag multiplies force by speed, so it grows approximately with speed cubed when the other quantities stay fixed.

That is why doubling speed can demand far more than twice the aerodynamic power. A larger gear can permit a higher speed at a given cadence, but it cannot remove air resistance.

Rolling resistance and surface

Rolling-resistance force is often modelled as a coefficient times normal force. The coefficient depends on tyres, pressure, surface and conditions. On a rough surface, a seemingly simple constant may be an incomplete description.

Rider physiology

Mechanical formulas cannot tell every rider the same ideal cadence. Comfort, training, fatigue, joint loading, bicycle fit and purpose differ. British Cycling offers practical cadence and climbing guidance for riders, but educational calculations here should not be mistaken for individual medical or training advice.

Safety and context

Students should conduct stationary or supervised measurements, away from traffic. Do not stare at a display while riding. A mathematically optimal gear is irrelevant if selecting it distracts from control, road awareness or safe braking.

Internal gears and other drivetrains

Not every bicycle exposes one chainring-to-sprocket ratio as the complete transmission. Internal-gear hubs, belt drives and electric-assist systems introduce additional stages or controls. The same reasoning still begins with rotations, but the overall ratio is the product of the ratios in each active stage. A student should use the manufacturer’s documented ratio for the selected internal gear rather than infer it from the visible sprockets alone.

Electric assistance does not change the definition of cadence or development. It changes how input power is shared and controlled. Range then depends on battery energy, assistance setting, terrain, speed, mass, wind, stops and efficiency. One distance claim cannot be guaranteed from gearing alone.

Chainline, wear and the limits of a one-number comparison

Two gear combinations can have almost the same ratio while placing the chain on different chainrings and sprockets. For instance, 50:25 and 34:17 are both exactly 2.0. The ideal development is therefore the same for a given wheel, but the physical arrangements are not identical. Chain angle, component size, available neighbouring gears and manufacturer recommendations can influence the sensible choice.

A ratio table should therefore be treated as a map of kinematic possibilities, not a maintenance manual. Cross-chaining advice depends on the drivetrain. Component inspection, adjustment and replacement should follow the bicycle and component maker’s instructions or qualified service guidance.

Wear also changes the real system gradually. A chain is made of links and joints; wear at many joints increases its measured length over several pitches. Measuring across many links reduces the relative influence of placing a ruler slightly off one pin. This is another example of why a longer measurement baseline can improve resolution.

Mathematics can help track change. If an initial measured reference length is 305.0 mm and a later measurement is 306.2 mm, the relative increase is (306.2 − 305.0) ÷ 305.0 × 100%, about 0.39%. Whether that calls for service is not decided by this invented example; use the correct tool, method and manufacturer limit. The calculation merely shows how a small dimensional change becomes a percentage.

This distinction protects against a common misuse of mathematical authority. A correct percentage does not create the threshold to which it should be compared. The threshold must come from an appropriate documented source.

Gear range

Gear range compares the largest available development with the smallest:

range percentage = highest ratio ÷ lowest ratio × 100%.

If the highest ratio is 4.0 and the lowest 1.0, the range is 400%. This means the highest ratio is four times the lowest; it does not mean there are four equally spaced gears. Range describes endpoints, while spacing describes the steps between them.

Both summaries matter when comparing how smoothly a rider can change effort.

Common misconceptions

“A higher gear always makes the bicycle faster”

Only if the rider can supply the cadence and power needed under the conditions. Speed depends on both gear and crank rate, while sustainable motion also depends on resistance and the rider.

“The smallest rear sprocket is the easiest gear”

With a given front chainring, a smaller rear sprocket produces a larger gear ratio and generally less torque multiplication. It is usually a harder gear, not an easier one.

“Cadence is wheel rpm”

Cadence refers to crank rpm. Wheel rpm equals cadence multiplied by the gear ratio while the drivetrain is engaged in the ideal model.

“A low gear reduces the energy needed to gain height”

It changes force and distance per pedal turn. The gravitational energy change for the same mass and height remains approximately mgh.

“A 10% ratio decrease means cadence rises exactly 10%”

At fixed speed, cadence varies inversely with ratio. Reciprocal percentage changes are not symmetric. Calculate the quotient rather than relying on intuition.

“Every labelled 700C wheel has the same circumference”

Actual rolling circumference depends on tyre dimensions, pressure, load and setup. Measure when accuracy matters.

“The formula proves which gear a person should use”

The formula describes relationships. It does not know the rider’s condition, traffic, surface, weather or goal.

How students can practise

Start with physical counting. Record the front and rear tooth counts for several combinations and compute the ratios. Arrange the gears from lowest to highest. Then predict which gives the greatest development.

Next, measure a wheel rollout and calculate development. Keep units beside every number. Use a spreadsheet to create a table of predicted speeds for several cadences. Graph cadence on the horizontal axis and speed on the vertical axis for different gears; each ideal relationship should be linear through the origin.

Then test sensitivity. Change circumference by 1%, cadence by 1% or tooth ratio by 1%. Because the speed formula multiplies these factors, a 1% change in any one factor creates approximately a 1% speed change if the others remain fixed.

Try an inverse problem. Choose a target road speed and cadence, then calculate the required gear ratio. Compare it with the ratios actually available. This introduces discrete optimisation: a bicycle offers a finite set, not every possible real number.

Finally, write a limitations paragraph. Mention measurement uncertainty, tyre rollout, freewheeling, drivetrain loss, drag and rider variability. A complete mathematical model includes the boundary of its own usefulness.

A compact investigation plan

Ask a precise question, such as how rear sprocket size changes development.
Record tooth counts and measured circumference.
Predict results before collecting motion data.
Use a safe stationary trainer or supervised setup if timing rotations.
Repeat observations and compare spread.
Explain discrepancies without forcing them to disappear.

This cycle—question, model, measure, compare, revise—is central to science, engineering and responsible data work.

Guidance for parents

A bicycle can turn a worksheet topic into a shared investigation. Ask your child to explain what one crank revolution does. Let them estimate before calculating. The explanation often reveals more understanding than the final decimal.

Use accessible questions: Why does the bicycle feel easier after shifting? Why does the same cadence create different speeds? Why might a phone and wheel sensor disagree? Each question invites ratios, units and evidence without turning the activity into a test.

Praise careful assumptions. A student who says “about 20 km/h in the ideal model” is reasoning better than one who reports an implausibly exact answer. Encourage diagrams and labelled units, especially when algebra feels abstract.

Keep the activity safe. Tooth counting, rollout measurement and spreadsheet modelling can be done with the bicycle stationary. Any riding should follow ordinary supervision, equipment and traffic-safety expectations.

For broader connections, compare this mechanism with mechanical clocks, gear trains and pendulum periods. The objects are different, but both use repeated cycles and integer tooth counts to control motion.

Frequently asked questions

Why is mathematics important in bicycle gearing?

It connects tooth counts, wheel size and cadence to development, speed, torque and climbing. It also helps a rider or designer compare trade-offs rather than relying on labels such as “easy” and “hard”.

What is the simplest bicycle gear formula?

For a derailleur bicycle, the ideal rotation ratio is front chainring teeth divided by rear sprocket teeth. Multiply that ratio by rolling circumference to estimate distance per crank revolution.

Does a bigger chainring make pedalling harder?

With the same rear sprocket, it raises the gear ratio and reduces ideal torque multiplication. Whether it feels harder also depends on cadence, speed, slope, wind and the rider.

What does cadence mean?

Cadence is the rate at which the cranks turn, usually in revolutions per minute. It is not automatically the same as wheel rotational rate.

How do I calculate bicycle speed from cadence?

Multiply cadence by wheel circumference and the front-to-rear tooth ratio to obtain distance per minute, then convert units to km/h or m/s.

Why is circumference better measured than assumed?

The loaded tyre’s actual rollout can differ from a nominal size because of tyre construction, pressure, load and surface. Direct measurement improves the model.

Is mechanical advantage free energy?

No. A lower gear can reduce required force while increasing pedal turns and time. Ideal machines exchange force for distance; real machines also have losses.

Can mathematics select the perfect gear?

It can identify ratios consistent with a target speed and cadence. It cannot determine a universal perfect choice because riders, conditions and purposes vary.

Is this topic useful beyond cycling?

Yes. It develops proportional reasoning, unit conversion, angular motion, energy, power, measurement and model criticism. Those ideas appear across physics, engineering, computing and daily decisions.

What should a student remember most?

Keep the quantities separate, carry units, state assumptions and interpret the result. A number without its physical meaning is not a finished answer.

Useful next reading

For a practical official explanation of tooth ratios and gear choice, read British Cycling’s Understanding gears and Gear Selection. Shimano’s official technical document library shows how component makers publish specifications and manuals; always match a document to the actual model.

Continue the mathematical journey with sports statistics, speed and performance for fair comparison, road curves, superelevation and stopping sight distance for transport geometry, and the Mathematics Learning Hub for wider study pathways.

Final perspective

Bicycle gears make the importance of mathematics visible in every rotation. Ratios translate teeth into wheel turns. Circumference translates turns into distance. Cadence translates distance per turn into speed. Torque and power explain why a change that helps on a climb may limit distance per revolution.

The deeper lesson is not that one gear is best. It is that mathematics lets us describe a trade-off, test a prediction and recognise the model’s limits. That habit—clear quantities, coherent units, checked calculations and honest interpretation—is useful far beyond the bicycle path.

Why Mathematics? | Bicycle Gears, Cadence and Mechanical Advantage