# Degrees and Radians

In everyday life, angles are usually measured in degrees: a right angle is 90°, an equilateral triangle has three 60° angles, a whole circle is 360°. The number 360 is nice because it is divisible by so many other numbers.Mathematicians and physicists, however, often like to measure angles in radians, rather than degrees. A radian is defined so that $$360^\circ = 2\pi \u{rad} \quad \hbox{or} \quad 1\u{rad} \approx 57.3^\circ$$ 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 $s={2\pi\over 6}R$ or $\frac13\pi R$.

But what is the angle $\theta$ of that slice? Well, if a complete circle is $2\pi\u{rad}$ around, then one-sixth of the way around should be $2\pi\over 6={\pi\over 3}$, 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 $\u{rad}.^\ast$ 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.