TITLE: Why do we have 360 degrees in a circle and why we need radians?
QUESTION [5 upvotes]: I have two related questions:
1- Why do we have 360 degrees in a circle? 
2- I have seen in most of the mathematical concepts, angle is expressed in radians not 
in degrees. Why was radian introduced?

REPLY [3 votes]: Possibly the most important reason for using radians instead of degrees to measure angles is that it's the simplest way for computing the straight length subtended by a circular arc: if we have a circle of radius $r$, then we know that the length of the arc subtended by an angle $\theta$ measured in radians is just $r\times\theta$ - a relation that doesn't hold true if we measure $\theta$ in any other units.  This in turn leads to a lot of observations that have important uses in 'higher' mathematics; for instance, the idea that a small piece of circular arc is approximately straight can be formalized as the statement that $\sin\theta\approx \theta$ for angles $\theta$ near $0$ - but only if we measure $\theta$ in radians!  A ton of important formulas can then be derived from versions of this formula: for instance, it yields the formula $\dfrac{d}{d\theta}\sin\theta=\cos\theta$, which doesn't hold true if we measure $\theta$ in degrees (it's off by a constant factor), and the list goes on quite a bit.  Radians are just, in an important sense, the most natural scaling factor for talking about angles.