Degrees and Radians

Mathematicians and physicists, however, often like to measure angles in radians, rather than degrees. A radian is defined so that $${360}^{\circ }=2\pi \mathrm{rad}\text{or}1\mathrm{rad}\approx {57.3}^{\circ }$$ Why would we ever want to use a unit as clumsy as that?

For one very good reason. A circle with radius $R$ has a circumference of $s=2\pi R$. Now consider a semicircle with the same radius: the length of the curved part of the semicircle is half of this, or $s=\pi R$. (The "length of the curved part" is called the arc length of the semicircle.) If we cut the circle into sixths, the arc-length of one of those slices is $s=\frac{2\pi }{6}R$ or $\frac{1}{3}\pi R$.

But what is the angle $\theta$ of that slice? Well, if a complete circle is $2\pi \mathrm{rad}$ around, then one-sixth of the way around should be $\frac{2\pi }{6=\frac{\pi }{3}}$, which is exactly the coefficient in the arc-length expression. This would work no matter how we slice the circle, and so we have the expression

There are some cases where knowing the arc-length could be very useful indeed; for instance, if you're thinking about the distance a car travels around a curve. Therefore, while we will usually use degrees when talking about angles, occasionally we will need to break out the radians. How can you tell the difference? Degrees will always have a ° symbol after their name, for one, while radians will often be tagged with $\mathrm{rad}{.}^{\ast }$ Generally speaking, if you see a $\pi$ in an angle, then it's probably in radians, although that's not a guarantee, and the reverse is less often true.