It’s much simpler than that.

NEW CRANKS DIVIDED BY OLD CRANKS, MULTIPLIED BY SPEED.

If you accept the basic idea that your feet move at the same speed, then shorter cranks mean your feet can do more revolutions per second, whilst moving at the same speed.

If your cranks are half as long, your feet have half as far to go, so they can do twice as many rpm.

If you do twice as many rpm, then you go twice as fast.

So, say you went from 6 inch cranks to 3 inch cranks, you’d be halving your cranks and doubling your speed.

6/3 is the same as 2/1. (Just do 6 divided by 3 = and you will get the answer 2)

So, going from 170 top 145, you use the same formula: original cranks divided by new cranks:

170 divided by 145 = 1.17

Your new speed will be 1.17 x whatever it was before.

Or, if you went the other way, 145 divided by 170 = 0.85, so cahnging back makes your speed 0.85 x whatever it is on the shorter cranks.

So, to calculate the approximate change of speed:

NEW CRANKS DIVIDED BY OLD CRANKS, MULTIPLIED BY SPEED.

That’s all there is to the maths.

But what is speed? Maximum speed? Cruising speed? Average riding speed over a mixed terrain journey? Shorter cranks will not affect all of these identically, because other factors like skill, control, acceleration/deceleration, and nerve come into it.

The “constant footspeed hypothesis” is a jokey name we give to a rough and ready rule of thumb. I have ridden wheels sized 20, 24, 26, 28 and 36, and have used various lengths of cranks including 89, 102, 110, 125, 150, 170, and have put some effort into making comparisons. The constant footspeed hypothesis is no more than an approximation, and only really works for adjacent crank lengths (say 125 and 150) with all other things being equal (wheel, tyre, terrain, rider etc.) and when the rider is confident and in control.

The rider and cranks are the engine. The wheel is the gear. Choose the cranks to suit the rider (we are all different sizes) and the wheel to suit the speed or terrain. A big wheel is the way to go fast, a small wheel is the way to go up hill.