What even is a "radian"?

When I first encountered the radian it seemed that the world had just become unnecessarily more complicated.  What was wrong with plain old degrees, and how did I know when to use which?  I found out it really helps to learn what a radian actually is.

Consider a circle of radius r, and imagine wanting to know the length of an arc, s, subtended by an angle θ.



Suppose the angle is such that s = r.  That angle actually defines the unit radian, so in that case θ = 1 radian.  Obviously, the arc length is proportional to the angle, so we get the general formula 

s = rθ

so long as we measure the angle in radians.  That's what makes the radian so useful: it gives a direct relationship between angular distance and length.  One place that has obvious applications is in rotational kinematics, because it allows us to easily relate angular velocity to tangential velocity.

But how big is a radian in degrees?  The circumference of a circle is the arc length for the whole circle (360 degrees).  In that case

s = 2πr

so 

 2π radians = 360 degrees

1 radian = 57.3 degrees (approximately).

In fact, you can now see where that famous formula relating the circumference of a circle to its radius actually comes from.  Imagine two ancient mathematicians discussing their circles:

Ancient mathematician 1: How many radians in a semi-circle?

Ancient mathematician 2: Dunno, but let's call it π.

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