Question

What makes the radian unit has a better approximation than 360 does?

Answer 1

The radian is a better measure than degrees for angles because:

1. It makes you sound more sophisticated if you talk in terms of irrational numbers.
2. It allows you to easily calculate the arc length without resorting to trigonometric functions.

(Point 2, is perhaps valid... point 1, no so much).

To a certain extent it is a matter of audience familiarity; where I live, if I were giving directions and told someone to go ahead 100 meters then turn right `pi/4` I would get some pretty strange looks in response ("turn right `45^@`" would be accepted as understandable without comment).

Answer 2

My personal idea is that:
The radian unit expresses the measure of an arc's length along the conference. This measure seems very concrete. For example: `2piR`
The degree reflects the measure of an angle starting from the center of the circle. This measure is of course very abstrait.