Let's briefly imagine some new, simple measure for seeing where we are on the circumference a circle. We'll call this unit "rotations," and we'll define it like this:
Let r be any number. r describes the number of rotations we've made around the circle. If r = 1, we've gone all the way around; if r = 1/2, we've gone half way around, if r = 1/4, we've gone a quarter of the way around, etc.
Once we've established that unit, we can use it to establish a link between degrees, radians, and arc length.
1 full rotation corresponds to:
- 360°
- 2π radians
- 2πr in arc-length (where r is the radius of the circle)
If we wanted to find unit conversions between each, we could just set up some equalities between the three:
Radians → degrees:
2π rad = 360°
1 rad = (180/π)°
Degrees → radians
360° = 2π rad
1° = π/180 rad
Radians → arc-length
2π rad = 2πr arc-length
1 rad = r arc-length
Arc-length → radians
2πr arc-length = 2π rad
1 arc-length = 1/r rad
Arc-length → degrees
2πr arc-length = 360°
1 = 180/πr arc-length
Degrees → arc-length
360° = 2πr arc-length
1° = πr/180 arc-length