As a mechanical engineer who has setup and solved many rotational systems, tau is a laughably naive idea and is utterly and completely wrong.

The discussion here is on choosing between pi, which is a radius-defined value, and tau, which is a diameter-defined value. Let's examine some definitions:

**Radians**The word radian is used to describe an angle as a ratio of the length of the circular arc to its radius. One radian is when the circular length is equal to the radius

With tau, we would have to call them diametans, and would now be the ratio of the length of the circular arc to its diameter. Now you might be saying that we could just draw the diameter in the picture above, because it is already a circle, but what about this next picture:

There is no circle. The radius lies along the arms, with endpoints at the vertex and the intersection of the arms with the arc. The diameter is completely irrelevant, and should not be considered.

**Angular Velocity (rad/s)**If something is spinning, it has a nonzero angular velocity. First, let's examine a case where both pi and tau will work: a wheel rolling on the ground without slipping. If I want to know how fast the wheel is rotating, I simply multiply the angular velocity by the radius of the wheel. For a wheel of radius 0.1 meters rotating at 20 rad/s, I can find that it travels at 2 m/s. In this situation, because the wheel is radially symmetric, I can do the same thing with tau diametrans. For a wheel of diameter 0.2 meters rotating at 10 diam/s, I can find that it travels at 2 m/s.

What happens when the object does not have a radius? Let's say the rotating object is the arm of a baseball pitcher as the throws the baseball, and we want to find out how fast he throws it. If we measure the length from his shoulder to the palm of his hand, then we know the radius of rotation. Let's imagine we did this measurement and got 0.65 m, thus when we see that he rotates a quarter of a circle in 0.1 seconds, that is 5pi rad/s, so we know that he can throw the ball at 5pi * 0.65 = 10.21 m/s.

Since a diameter is not always defined in the system, it would be incorrect to define the angle based on the diameter.

**Torque (Nm)**Torque is the equivalent of force for rotational systems. The first thing to note right away is that equations already use the tau symbol to represent torque, so tau should fuck off and find a different name. Torque is defined as a force acting at a radius from a center of mass, orthogonal (perpendicular) to that radius. Yes, torque is defined based on radius, and can't be defined based on diameter. There is no such thing as a diameter to a center of mass, it literally doesn't exist. In fact, as you go further along in your math career, you will find that diameters are only a part of science and engineering equations, and are dropped entirely from the mathematics curriculum, because they are not an independent value, but are a function of the radius, and can be undefined in many cases.

**tl;dr: tau is a terrible idea, pi makes sense, radians are a ratio, not an arbitrary number that you can fuck around with.**