A complete revolution is 2π radians. It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.
It follows, via observation, that the 'Radian' relates to pi in a unique way ...
“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.