“As the ratio of two lengths, the radian is a “pure number” that needs no unit symbol.”
The degree is also similarly mathematically derived. Whether we assign it a unit symbol or not is really purely a subjective decision by humans.
“It follows, via observation, that the ‘Radian’ relates to pi in a unique way ...”
Well, really, it follows that the ratio of a circle’s radius to its circumference is a unique and defining characteristic of a circle. Pi is what is unique, radians are just a unit that we invented to measure angles in relation to that ratio, rather than any other mathematical relationship we might have chosen.
re: “The degree is also similarly mathematically derived.”
Nop. Artificial, arbitrary selection of value.
You’re just not getting it. Maybe some day, you will ...
The “Degree’ (angle measure)
From wiki -
The original motivation for choosing the degree as a unit of rotations and angles is unknown.
One theory states that it is related to the fact that 360 is approximately the number of days in a year.
Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers.
Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree.
Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.[11] Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.
The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda