
This free javascript bicycle chain length calculator uses the "rigorous equation" as published on the Park Tools site and various discussion forums.
The "rigorous equation" takes the following form:
TSPCL = 0.25 * (F + R) + 2 * SquareRoot( [CSL * CSL] + {0.0796 * [F  R]} ^2 )
Where:
TSPCL = Theoretical Shortest Possible Chain Length
CSL = Chainstay length in inches
F = Front gear cog teeth
R = Front gear cog teeth
I have no idea how the magic number of 0.0796 was derived but the formula appears to provide a pretty good approximation to the shortest possible length of 1/2 inch pitch bicycle chain that will theoretically go around the two cogs on a fixed wheel bike. The value is then rounded up or down to the nearest whole link value (nearest inch), or up if the unrounded value is exactly xx.5. Because you obviously can't have fractional parts of a chain link, or gear cog tooth.
Many modern derailleur geared bikes have vertical frame dropouts that don't permit the back wheel to be moved. So rounding downwards is a bad idea. One link too long is not much of a problem, but 1/2 a link too short can be a real pain! If you have one of these select 'Fixed chainstay length' and the value will always be rounded UP instead of up or down to the closest whole inch.
For derailleurs one inch is added to TPSL to allow for the extra length of chain to go around the derailleur jockey wheels when the biggest front cog is being used in conjunction with the biggest rear cog. Most rear derailleur mechanisms have two ten tooth jockey wheels and an inch will be fine for these. I have seen some mechanisms with 15 and 13 tooth jockey wheels though and I would mistrust the add an inch rule with these adding 2 links instead.
NOTE: This chain length calculator is only valid for 1/2 inch pitch chain  The standard pitch for bicycles
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