Notice that there is a nice linear relationship between the angular position of the wheel and the horizontal position? The slope of this line is 0.006 meters per degree. If you had a wheel with a bigger radius, it would move a greater distance for each rotation—so it seems clear that this slope has something to do with the radius of the wheel. Let’s write this as the following expression.

In this equation, *s* is the distance the center of the wheel moves. The radius is *r* and the angular position is θ. That just leaves *k*—this is just a proportionality constant. Since *s* vs. θ is a linear function, *kr* must be the slope of that line. I already know the value of this slope and I can measure the radius of the wheel to be 0.342 meters. With that, I have a *k* value of 0.0175439 with units of 1/degree.

Big deal, right? No, it is. Check this out. What happens if you multiply the value of *k* by 180 degrees? For my value of *k*, I get 3.15789. Yes, that is indeed VERY close to the value of pi = 3.1415…(at least that’s the first 5 digits of pi). This *k* is a way to convert from angular units of degrees to a better unit to measure angles—we call this new unit the radian. If the wheel angle is measured in radians, *k* is equal to 1 and you get the following lovely relationship.

This equation has two things that are important. First, there’s technically a pi in there since the angle is in radians (yay for Pi Day). Second, this is how a train stays on the track. Seriously.