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Math in Motion :: The Bicycle . Jason Achilich GED 613 Math Notebook. Let’s Start with the Frame . There are many examples of math in bicycle frames. The most basic is the size, Size is measured from the center of the cranks to the top of the seat tube. What about the reach of the bike?.

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math in motion the bicycle

Math in Motion ::The Bicycle

Jason Achilich

GED 613

Math Notebook

let s start with the frame
Let’s Start with the Frame
  • There are many examples of math in bicycle frames.
  • The most basic is the size,
    • Size is measured from the center of the cranks to the top of the seat tube.
what about the reach of the bike
What about the reach of the bike?

Now that we see how the height of the bike is measured.

What about the reach?

The top tube is another important number to fitting bike frames.

This number is proportional to the seat tube, as one gets larger, the other measurement grows as well.

angles in a frame
Angles in a Frame

Yes there are angles in bicycle frames.

Both the seat tube (ST) and the head tube (HT) are angles.

These angles affect how the rider is positioned on the bike and how the bicycle handles.

math in bike wheels
Math in Bike Wheels

Sometimes different wheel sizes are used for performance, other times to go faster, sometimes to fit a smaller person on a bike.

Generally a very short person will have smaller wheels on their bike to make positioning easier.

how does wheel size vary with height
How does wheel size vary with height?
  • You can see in this picture that a shorter person rides a bike with proportionally smaller wheels than a larger person.
a larger mountain bike wheel
A larger mountain bike wheel
  • Over the past few years, some mountain bikes have started to have larger wheels.
  • Why?
  • The larger size wheel makes objects in the trail feel smaller, since the wheel hitting the object is proportionally larger.
  • This makes rolling over objects easier since the object is hitting lower down on the wheel.
spokes in the wheel
Spokes in the wheel


  • All wheels are built to a certain tension on the spokes to hold them together. When a wheel has more spokes the tension is distributed across them all, resulting in lower tension on each spoke.
  • Many bike also come with wheels that have very few spokes. Yes you guessed right, less number of spokes equals more tension or force on each spoke, since there is less distribution of force.

Does it matter?

  • Yes it does, it helps to create a stiffer wheel, plus there are fewer spokes to hit the wind. But also the greater force on each spoke requires more work to be done by each spoke to keep the system neutral. If one of them break, the system is greatly out of balance.

Wheels for different styles of cycling sometimes have different numbers of spokes.

The more spokes that a wheel has the stronger it is.

can a wheel not be round
Can a Wheel not be Round?

A wheel will not be oval shaped, but it is possible for a wheel to not be perfectly round.

When viewed closely, there can be high and low spots.

These hops and dips are measured in millimeters, and though these small variations do not affect the ride, they can lead to imbalances in the system.

penny farthings
Penny Farthings

Penny what? Penny Farthing, a term based upon two British coins of the time, a Penny, and a Farthing which is a quarter Penny. The two coins resemble the bicycle.

These were the first bicycles, dating back to the 1860’s. At this point there were no gears to make a rider go faster or slower.

James Starley found that with a direct drive bicycle, a larger wheel could travel further each rotation because of the larger circumference of the wheel.

examples of penny farthings
Examples of Penny Farthings
  • The diameter of the drive wheel on this bike is 40 inches.
  • While this drive wheel is 58 inches in diameter.

Which bike will go faster when pedaled at the same cadence?

making the wheels turn
Making the Wheels Turn

Lets look at the gears:

  • This would be a hard gear: for every one rotation the front chainwheel makes, the rear makes four rotations.
  • This gear is very easy; it would take many rotations of the front chain wheel to move the rear gear once. Very slow, but very easy to climb big hills with.

Unlike the Penny Farthings, modern bicycles have an assortment of gear ratios. This range of ratios allows the rider to work more or less, depending on how fast they would like to travel.

using the gear ratios
Using the Gear Ratios
  • Compared to racers “against the clock” use a higher gear to keep a constant speed.

Track racers use ratios to their advantage.

Sprinters need quick acceleration, by the use of a lower gear, to get a jump on the competitor.

Putting their work into a lower gear allows the whole system to increase in speed quickly.

pedaling the bike
Pedaling the bike

The length of the crank arm also is affected by math.

A longer crank are will give the rider more leverage to push the gear, but the circle their feet must travel is larger, causing a slower rotational time.

While some riders will use a shorter crank length, giving them less leverage, but a quicker cadence.

The eternal question, which is more


numbers where bikes are ridden
Numbers where bikes are ridden.

Track racers compete on an elliptical course called a velodrome.

These velodromes can range in length from 250 meter to 400 meters.

A 250 meter long track can have a banking of 50 degrees in the two turns.

A 400 meter track has banking of 30 degrees

Longer turn equals less need for banking.

riders using the banking
Riders using the banking

Track racers will use the banking to their advantage.

To gain an advantage a racer will get above their competitor on the banking.

Once it is time to sprint, they will use the slope of banking to help them accelerate.

Here a rider gets up on the banking to accelerate out of the turn.

traveling around the velodrome
Traveling around the Velodrome

In track racing much of the winning and losing depends on your line around the track.

The lines on the track represent “lanes.”

Riders can’t ride below the black line near the bottom, so riders will hug it, enabling them to ride a shorter distance than the rider trying to “come around them.”

how many kilometers in an hour
How many Kilometers in an Hour?
  • How many kilometers did he ride?
  • The current hour record holder rode an average speed of 49.7 Km/Hr in 2005.

The hour record on the track is the purest form of cycling ability.

It is the race against the clock.

Competitors attempting to break the hour record will know the average Km/Hr that they will need to ride.

The rider shown here averaged 49.431 Km/Hr in 1972 to set the new standard.

meters what s that in miles
Meters, what’s that in miles?

Cycling roots are very European, so most all events are measured in metric units.

Many cycling fans that are not using the metric system are familiar with the kilometers to miles conversion.

It is common for a cycling fan to know some key conversions.

One kilometer is about a six tenth of a mile and one mile is about 1.6 kilometers.

Which allows a rough estimate of a large European road race of 200km, to about 125 miles.

percent grades
Percent Grades





Since cyclists are powering themselves, they worry about the steepness of the hills they are riding.

This steepness is generally represented in percent grades.

To find the grade, divide the rise, by the run, and multiple it all by 100.

A mellow grade is below 5 percent.

22 percent is the steepest grades found in Vermont

route planning
Route Planning

This spring I am participating in the New England Fleche.

A Fleche is a ride where there is an ending point, but the starting point can vary; teams build their own route to the finish around specific rules regarding distance and time.

The main rule requires teams to ride at least 380km in a 24 hour period. What will the average speed be for this criteria?

My team has put together a spread sheet adding up our mileage and times. Each segment leg is added to our total, so we know how far we will traveled in 24 hours.

Our time is also added up by segments to form the whole. Thus ensuring we follow the rules.

our trip to westfield ma
Our Trip to Westfield, MA
  • You can see how we have added the mileage in each leg to form the total mileage we will be covering.
rider position on the bicycle
Rider position on the Bicycle

A rider can benefit from proper positioning on a bicycle. A right angle, is also the strongest.

In this photo you can see all the different angles in play.

It is important to notice the

angles on the torso.

She is using structure not

muscle to hold herself up.

handlebar measurements
Handlebar Measurements

Comfort on a bike can even be affected by your handlebar width.

Too wide and you are using strength to hold yourself up.

The right width you are able to use

a 90 degree angle to help support

your weight.


In my searches for information concerning math in cycling, many websites devoted to the topic were uncovered.

The information presented here barely scratches the surface of the topic, but what is presented are some of my favorites that I use and think about most often.

Math is everywhere in cycling, looking at the sport it is staggering how in-depth you could take many of these examples!

onward a final thought on pi
Onward:A final thought on Pi.

The meaning of Pi now has meaning to me, as I almost daily use it to determine the circumference of a wheel while installing a cyclo-computer.

I am looking forward to exploring more cycling and math relationships down the road.