Question

Why are there 2 pi radians in a circle?

Answer 1

It is not arbitrary, but it is because of the definition of radian measure.

Explanation:

One definition says the radian measure of angle `theta` is the ratio of arc length `s` to radius `r` when `theta` is made central in a circle of radius `r`.

The arc length of the whole circle is its circumference `2pir`, so the angle that goes once around the circle has radian measure `C/r = (2pir)/r = 2pi`

This definition is the reason that we have the formula for arc length `s = r theta` as long as we measure `theta` in radians

Note that whatever unit are used to measure `r` and `s`, in the definition of radian measure they cancel. We say that radian measure is dimesionless.

Also note that some teachers introduce radian measure without discussing the 'official' definition as a ratio of lengths. They simply announce that once around the circle is `2pi` radians. (Some say `360^@` is the same as `2pi` radians.)
This makes it look arbitrary when it is not arbitrary at all.