… I meant pi, not 2. At least in the first formula.
Aaaargh.
I guess this would be a good time to ask the question. I want to get a 20" trials uni and based on that chart a 125mm crank arm would be a good choice because it gives me good speed AND control over my uni?
I ride 175 cranks on a bike, but on a uni, it feels like too much throw. anyway, the wierd thing is I use 150s on my 26, 29, & 36er (sometimes 125 on 36er)! 150’s just seem right all around even on different wheel sizes. I don’t ride 20", but if I did, I would expect I’d ride 135/137mm cranks. I don’t understand why I like the 150’s oddly enough. 
BlueFlyer83:
A trials would be comparable to a 24" Muni w/ 165 or 170’s. You want the length for control while on obstacles and you don’t care about rolling speed. You don’t worry as much about pedal strikes since you mostly hop on/over the obstacles instead of rolling through like on a Muni. So go w/ 135-140.
If you are going to do street or flat on your trials you may want shorter cranks to get speed for jumping stairs and faster flips. If that’s the case you may prefer 114-125. Plenty like the same longer length as “regular” trials. I’ve seen street riders get tons of speed for jumping stairs (8 set) w/ 145 cranks on their trials uni.
Thanks for the advice!
good thread.
Just added 27.5 wheel size to Eric’s creation. I wasn’t aware that he had already posted the 32" wheel… to be honest that was my personal driver to do this.
I was a tad too lazy to colorize all of his categories… This version has fewer bands of colors.
In case the image refuses to post:
https://www.dropbox.com/s/a0zor7aab18t8k2/Uni%20Crank%20v%20Wheelsize.GIF?dl=0
The colour bands are only for guidance and are very subjective anyway. So many variables affect the ideal ratio.
Off course it is only for guidance and subjective. But I find it pretty much sums up what you would get from a: “what cranks should I ride on a XX” unicycle?" thread in this forum, all in one compact picture. Which is rather nice, I think.
140mm cranks on a 20" puts us in the “pedal strike” zone, according to the chart. A beginner wondering which cranks to get on their 20" might avoid 140mm because of this chart. I have 140s on my trials unicycle. I am able to force pedal strikes by riding in fairly tight circles, but for everything else, I almost never have pedal strikes. The benefits of the longer cranks for me outweigh the occasional pedal strike (which almost never causes a UPD, anyway).
Notice that Biped kept the ratios for the descriptors, just halved the number of colours. 140 on a 20" wheel is still in the trials range with a gain ratio of 1.81. Depending on the terrain you can pedal strike with nearly any combo, but in my experience striking obstacles offroad with 150s on a 20" is pretty much a sure thing, hence switching categories somewhere between those sizes.
I used to switch things up a lot, and my experience with a lot of the combos is what I based the categories on at the time. My preferences have shifted somewhat over time, but as has been pointed out a few times, it’s just meant to be used as a rough and easy to use guide.
I am taking part in a WR attempt to try and break the 130 year old one hour record for Penny Farthings. These calculations of course work for Penny farthings as well as unicycles.
It is being done in pretty controlled conditions in a velodrome. For that I am riding a 50" wheel and have just switched from 100 to 90mm cranks and… yes faster and just about able to push. Can’t stop and it takes ages to wind up to speed without the banking to help.
Roger
This clearly needs a dedicated thread! We need to be kept up to date on this.
Tell us more about the existing record. We need to have pictures and videos of the machine you will be using at the very least.
Let me guess; it will be super-aero, with fairings and the rider mounted behind the wheel for better aerodynamics, right? 
Or not. Best of luck breaking that old record! Hopefully it will stand at least as long as the current one.
A recumbent penny with a 7.0 gain ratio? Fittingly that goes right past the “Danger Will Robinson, Danger” category and into the area I would not even consider humanly possible.
But then I’ve seen Roger riding a 36 with 100mm cranks on crumbling pavement while reading a book as others were just trying to keep up.
I thought I saw a post on facebook with Roger’s record attempt cycle, looks like and “ordinary bike” to me, there really should be a separate thread for this 
ok… will do a separate thread. Not that long until the attempt. Will try and get it up today. It is all very exciting. We had a velodrome practice this weekend and it is fun going fast in such safe environments.