Radians

Why does a full circle have 360 degrees and not 100 or 1000? The number comes from Babylon and is a practical agreement, because 360 can be divided by many numbers without a remainder. Mathematics does not force it on us. There is a second way to measure angles that needs no such agreement at all: you measure how far you have walked along the edge of the circle. That is radian measure, and in higher mathematics it is the default.

The angle as an arc length

Take the unit circle, the circle with radius 1. A point starts at the far right and travels counterclockwise along the edge. The distance it covers is the angle in radians. If it walks exactly a distance of 1 along the edge, it has swept out an angle of 1, spoken as one radian. That is roughly 57 degrees, a little less than a sixth of a full turn.

Why a full turn is 2π

A full turn means walking all the way around the edge once, so exactly the circumference. For the unit circle the circumference is $C = 2 \cdot \pi \cdot 1 = 2\pi \approx 6.28$. That is why a full turn in radians is $2\pi$ and not 360. Half a turn is therefore $\pi \approx 3.14$, a quarter turn $\frac{\pi}{2} \approx 1.57$. Rearranging $b = \alpha \cdot r$ gives $\alpha = \frac{b}{r}$: a length divided by a length, so a plain number with no unit. The label rad is only added as a reminder that the number is an angle in radians.

Converting between degrees and radians

Both measures describe the same full turn: 360° corresponds to $2\pi$. Divide both by 2 and you get the anchor worth memorising: 180° corresponds to $\pi$. So from degrees to radians you multiply by $\frac{\pi}{180}$, and the other way round by $\frac{180}{\pi}$. An example: $60° \cdot \frac{\pi}{180} = \frac{\pi}{3}$, because 60 fits into 180 exactly three times. Conversely $\frac{\pi}{4}$ is a quarter of 180°, so 45°. This is where the values you meet all the time come from: 30° is $\frac{\pi}{6}$, 45° is $\frac{\pi}{4}$, 60° is $\frac{\pi}{3}$, 90° is $\frac{\pi}{2}$, 180° is $\pi$ and 270° is $\frac{3\pi}{2}$.