 Mathematics
15 August, 10:32

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

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1. 15 August, 10:45
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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°

- 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:

Arc-length → degrees

2πr arc-length = 360°

1 = 180/πr arc-length

Degrees → arc-length

360° = 2πr arc-length

1° = πr/180 arc-length