Some while ago, I covered how to convert degrees into radians (and vice versa) in your head. I missed a trick, though: I didn’t tell you about the exact values, which would probably have been a bit more useful.

By definition, a circle – 360º – is 2π radians, which (hopefully obviously) means that a semicircle – 180º, keep up – is π radians. That’s a handy thing to know.

Getting into degrees

This is the wrong way! As I’ve said before, degrees are far inferior to radians in every possible application. But still, I imagine you might want to go back to baby-angles once in a while.

You’ll quite often see an angle given in radians as some fraction of π – π/6 or 3π/2 or similar. It’s pretty easy to turn those into degrees: if you’re good with fractions, just replace the π with 180 and work it out: π/6 becomes 180/6 = 30º. 3π/2 becomes 3 × 180 / 2 = 3 × 90 = 270º ((If you prefer, you can do that as 540 / 2 = 270, but I prefer dividing first. Small numbers are easier to work with.)).

If you don’t like fractions, I’ll sit here and roll my eyes a bit, then tell you that if there’s a number on top of the fraction, you times by it; and then you divide by the number on the bottom. Meanwhile, I’ll line up a Teletubbies DVD for you or something. You’re an A-level student, you need to be able to work with fractions.

Reverting to radians

Ah, that’s better. Turning things into radians is, of course, the way forward. It’s a simple three-step process:

  1. Put 180 under your degree angle, making it a fraction – 300º becomes 300/180
  2. Cancel down your fraction (here, there’s a factor of 60): 5/3
  3. Put a π either on top or next to the fraction – you can write $\frac{5\pi}3$ or $\frac53 \pi$. ((Avoid, please, 5/6π. That’s technically correct, but can easily be misread as 5/(6π). If you have to fall back on ‘it’s technically correct’, you’re doing it wrong.))

And that’s it. Of course, it’s worth knowing the more common ones off the top of your head:

  • π/6 is 30º
  • π/4 is 45º
  • π/3 is 60º
  • π/2 is 90º

Thanks to @C_J_Smith for pointing out a mistake in the original version.