How to calculate Gear Ratios

1. Preface
2. Low Gear
3. High Gear
4. Other Calculations
5. Friction Comparison
6. Terminology Summary
7. Deraileur Capacity (Chain Slack)
8. Power Output

1. Preface
What range of gear ratios do you need for a touring bike? For any city with hills such as Vancouver, San Francisco, etc. You typically want lower gears than come with any bike. Gear ratios are probably one of the first configuration changes you would make to a touring bike. Many people buy bikes and think they've got sufficient range of gear ratios. So I created a little database which allows you to see the range of some actual touring bikes. Gear Ratio Calculations Database. You can do the same calculations on your own bike, and see how it compares.

When comparing bikes with the same size wheels, the simplest way to calculate the lowest gear is to just divide the number of teeth on the smallest front chain ring with the number of teeth on the largest rear cassette cog. This could be called "cog ratio". For example my current low gear bikes have a 42/32/22 front chainring, and a 11-34 cassette, so the lowest gear is 22/34 = 0.65. But a typical "Hybrid" bike might have a 48/38/28 and 11-28, giving you a 1 to 1 ratio. Which is not a low enough gear for a loaded bike on a hill.

The above cog ratio comparisons assume you are comparing two bikes with the same size wheels. If you want to compare two bikes with different sized wheels, you want to calculate something called "Gear inches". Just multiply each of the above "cog ratios" by the wheel diameter in inches. A typical 700c wheel is 27 inches in diameter. So the gear-inches of the above 0.65 ratio is 18 gear-inches. A 27" wheel bike with a 1:1 cog ratio is 27 gear inches. Gear Inches is how far you go with each pedal rotation. So if you only go 18" with each pedal rotation, that is a lower gear than if you go 27 inches.

Another ratio often quoted is the overall "gear range", which is the highest to the lowest. Eg: On the Trek 520, the highest gear is 48/11 and the lowest is 26/36.

                      High     Low      Range
           Fraction   Ratio    Ratio    Percent
------------------------------------------------
Trek 520   48/11       4.4     0.72      610%

2. Low Gear
(Lowest Gear by cog ratio) here are two main ways to compare low gears: (1) Cog ratio and (2) Gear Inches. Gear Inches takes into account the wheel diameter. The cog ratio is just the ratio of the cogs, without regard for the wheel size. "Low Gear" by cog ration is is the ratio of the smallest chain ring on the front and the largest cog on the back cassette. Front/Back. If you've got a 24 tooth chain ring on the front and a 36 tooth cog on the back, your low gear is 24/36 teeth =.66 to 1. One turn of the crank turns the wheel .66 of a revolution.

On touring bikes, you definitely want your low ratio to be less than 1 to 1. For example, the Surly Long Haul Trucker comes with a 50/39/30 on the front and a 11-36 on the back. So the lowest gear ratio is 30/36 which is 0.83. My Miyata 1000 has 50/40/28 and a 14-30 on the back, giving 0.93, and my Rocky Mountain is 42/32/22 and 12-32 giving 0.69.

Of course, the gear ratio alone doesn't quite tell the whole story, because it doesn't take into account the diameter of the wheel and tire. There is a measurement called "Gear Inches" which just multiplies the above ratio times the diameter of the tire. eg: A 1.00 cog ratio with a 26 inch wheel would be 26 gear-inches. That is a handy number to compare if you are trying to compare two radically different sizes of wheels, such as a foldup bike and a road bike. If you really want to be fancy, you could factor in the crank length, and make adjustments for the actual tire diameter you have.

As a practical matter, if you are looking at buying a bike, simply calculating the cog ratio is good enough. Below are a few low gear statistics:

                  --Cog Ratio------
   Bike           Fraction  Decimal   WheelD  Gear-Inches
-----------------------------------------------------
  Surly LHT       30/36     0.83       27"      22.4
  Miyata 1000     28/30     0.93       27"      24.2  
  Rocky Mountain  22/32     0.69       26"      17.9

3. High Gear
The gear ratio is a ratio between the number of teeth. Could be called "cog ratio". It is calculated by dividing the number of teeth on the largest chainring by the smallest cog on the cassette. Eg: 52 teeth on the front divided by 11 teeth on the back. 52/11=4.7 (4.7 to 1)

4. Other Calculations
There are numerous gear calculators on the internet that calculate all sorts of other statistics. I don't find many of them useful these days to help with actual buying decisions. However you may find them interesting. For example, Sheldon Brown has a more complex "Gain Ratio" calculator. Sheldon Brown Calculator But for my purposes, all I want is to compare the highest and lowest ratios, because we can assume all the gears in between are roughly spaced equally.

But the Sheldon Calculator was more useful back in the days when we used to individually build up cassettes with individual rings.

Another statistic is called "Rollout" or "Development". Sheldon Brown talks about. It is the distance the bicycle travels for one crank revolution, and is usually measured in meters. Development can be calculated by dividing the chain wheel size by the rear sprocket size, multiplying the result by the wheel diameter and by pi (3.1416). See also gear inches and gain ratios.

So the rollout for the 0.72 ratio we multiply by 700cm x 3.14159

I don't think this is very useful, although one could use it to compare riding to walking I suppose.

5. Friction Comparison
This website did a comparison of losses between a 1x and 2x system. [a href=https://www.velonews.com/gear/gear-issue-friction-differences-between-1x-and-2x-drivetrainsVelo News[/a]

The 2X drivetrain had lower frictional losses in every gear than the 1X system—with the caveat that the chainrings matched up with the ideal cogs in the rear. The average friction within the 1X drivetrain was 12.24 watts. This was computed as the sum of the drivetrain power losses in each of the 11 gears divided by 11. The average friction of the 2X drivetrain was 9.45 watts, computed as the sum of the power losses in each of the 15 optimal gears divided by 15. This is just under a three-watt average difference between the two drivetrains.

The frictional losses of the system were highest for each chainring when the chain ran on the smallest cog. This is also where the difference in frictional losses were greatest between the two drivetrains: at the highest gear ratio (4.8) the 48 X 10-tooth combination consumed six watts more than the 53 X 11-tooth combination.

6. Terminology Summary
In my database, the various statistics have short code names like "Front" or "Rear" that require longer names if you were referring to them in a conversation or article.

Calculator    Article
 Name         Name
--------------------------------------
  Front       FrontCogs                Front Chainrings
  Rear        RearCogs
  LowGear     Low Gear Cog Ratio
  HighGear    High Gear Cog Ratio
  LowInch     Gear Inches - Low Gear
  HighInch    Gear Inches - High Gear
  ChainS      Chain Slack (Derailleur capacity)

  A typical conversation might be: Compare the "LowGear" on the Miyata 1000 with the Trek 520.

1. Cog Ratio
    The ratio of the number of teeth between front and back. There are two cog ratios: Low Gear and High Gear.

2. Cassette Range
    This is the typical spec you see in online catalogs. Eg: 11-34 It is usually accompanied by the number of speeds. Bike X has an 11-34 9 speed cassette. Only the two ends of the range are specified.

3. Low, High
    These words always refer to gears. Eg: In a car you might say "put it into Low".

4. LowGear
    Low Gear Ratio. The ratio between the crank and the axle. In my stuff I usually mean Low Gear Ratio if I just say "Low gear". But if I was sending an email I might say "what is the low gear ratio on that bike? Or what is the internal low gear ratio on a Rohloff hub? (Reduction ratio)

5. ChainS
    Chain Slack. The difference in teeth between the most extreme ratios. Eg: 42/32/22 and 11-34 has a difference of 43 teeth. That requires the derailleur to take up that much slack. To do that, it needs to be a long arm derailleur. If you are converting an old road bike to something with lower gears, you often have to get a different derailleur.

7. Deraileur Capacity (Chain Slack)
Every rear deraileur has a maximum capacity to take up the slack as you shift through various gear ratios. The "capacity" is quoted as difference in tooth count as you go between the extremes. I sometimes call this capacity number "chain slack".

The "gear" that takes the most chain is when the chain is on the largest cog on the front and back. The gear that takes the least chain is when the chain is on the smallest cog front and back. That difference gives you a number of teeth which derailleur specs call "capacity". Eg: On my Rocky Mountain with the mountain bike standard 42/32/22 and 11-34 cassette, the teeth difference is 43 teeth as follows:
 

   42+34=76
   22+11=33
        ---
         43  

Extreme road bikes need very little derailleur capacity because there is not much range in cog sizes. But touring bikes require much more capacity because the cog sizes vary more. In general all touring bikes require Shimano SGS (long cage) deraileurs. SGS is the Shimano code for "long cage". Shimano MTB rear derailleurs last letters will be GS or SGS, which stand for "Short cage" or SGS for long cage. A XTR rear derailleur will be RD-M970-GS (for short cage) or RD-M970-SGS (for long cage)

Park Tools Rear Derailleur Adjustment
 This article goes thru how to set the rear derailleur all three screws: H, L and B.

8. Power Output
Why do you need low gears? Because you need to keep spinning in your efficient range. A 20 inch low gear is 0.508 meters forward. If on a 1% slope, that is 0.005 meters up. Raising 1 kg 1 meter/second takes 10 watts. Raising 1 kg

So if spinning at 80 rpm,