Bicycling - Connecting Gear Inches, Cadence (RPM) and MPH

The formula to calculate MPH given RPM (and Gear Inches and a few other things)

When riding fixed gear what you see is what you get. Or rather what you choose in gear ratio is what you get.  How fast you can comfortably go is directly related to the gear setup you ultimately decide on.

I am one of few fixie riders among many roadies and its important for me to choose the right gear.  Choose to low of a gear and I can't keep up with the group.  Choose to hard of a gear and I'm working to hard.  And add the variances in elevations given any ride and well its complicated.

I am always thinking about the perfect gear.  For me it's 46 x 15 or somewhere between a 3 to 3.07 gear ratio.  This translates to a max speed of approx. 29 mph at about 120 rpm.  120 rpm can be a bit uncomfortable but its manageable and most group riders can't sustain this for long.  At 90 to 100 RPM the max speed is between 21 to 25 mph.

I bought a very basic cyclo-computer for my fixie.  It tells me MPH but not RPM.  When I am working out on my rollers, I try to maintain a certain perceived effort.  Knowing my MPH, I often wondered what RPM or cadence this works out to be.  You see, I am more interested in RPM than MPH but my cyclo-computer only tells me MPH. Maintaining a certain RPM is much better than worrying about how fast I am going and they are directly related.

Whew, to much thinking.  So, knowing MPH you can calculate RPM and vice versa.  I wanted to understand this relationship better so I went online to find a formula.  Unfortunately, most of the formulas were incomplete or did a horrible job explaining it.  I found a number of online calculators that did the math for you but I was more interested in understanding the relationship. For me understanding this relationship (or mathematical representation) will helps me appreciate the importance of choosing the right gear and explains to me what is going on given a certain RPM. 

Below is the formula to determine your miles per hour (MPH) given your gear ratio, tire size and revolutions per minute (RPM).  Sheldon Brown has a great calculator to figure this out.  He even takes into consideration crank length!  Since I was very disappointing in all the other explanations, I hope my version is easier to understand.  

In high school we are taught Distance = Rate * Time.  Thus Rate = Distance / Time ... And so the story begins!

For The Example I am running a 48t Front Chain Ring; a 16t Rear Cog and my Cadence (RPM) is 90.  I am running a 700c X 23 road tires.

Front Chain Ring = 48 teeth
Rear Cog = 16 Teeth
RPM = 90
Tire Size = 700c x 23 (This translates to a tire diameter of approximately 26.3 inches)
Pi = 3.14159

Step One: Calculate Gear Inches (GI)
GI = (Cf / Cr) * Diameter of wheel
Where Cf = # of Teeth of Front Chain Ring; Cr = # of Teeth of Rear Cog
Example = (48 / 16) * 26.3 = 78.9

Step Two: Calculate distance wheel will travel in one revolution (Dr).  To do this you will need the circumference of tire.  To calculate this times GI by Pi.
Where Pi = 3.14159.....
Thus GI * PI or GI * 3.14159
Example = 78.9 * 3.14159 = 247.8715

Step Three:  Calculate the total distance traveled in one minute (Dm in).  To do this you will need to insert a given number of Revolutions Per Minute or RPM.  To calculate the distance traveled on on minute take your Dr and times it by your RPM.
Thus  (Dm in) = Dr * RPM
Example = 247.8715 * 90 = 22,308.435

Step Four: Convert (Dm in) in to feet (Dm ft). The number you get in Step Three is in inches. You need to convert it to feet to eventually determine MPH.  To do this simply take (Dm in) and divide by 12.  You divide by 12 because 12 inches equals one foot.
Thus (Dm ft) = (Dm in) / 12
Example = 22,308.435 / 12 = 1,859.03625

Step Five: Calculate Distance traveled in one hour (Dh).  Since there are 60 minutes in an hour you take the number in Step Four and times it by 60.
Thus (Dh) = Dm ft * 60
Example =  1,859.03625* 60 = 111,542.175

Step Six: Calculate Mile per Hour (MPH).  This is the final step.  The number you have in Step Five tells you your total distance traveled in one hour.  The number tells you total distance in feet.  There are 5280 feet in one mile.  to to convert to miles you simply divide the number in Step Five by 5280.
Thus MPH  = Dh / 5280
Example = 111,542.175 / 5280 = 21.125; Given my current gear setup and my desired RPM, I will be traveling at 21 mph.