Explain the relationship (s) among angle measure in degrees, angle measure in radians, and arc length

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