Secrets of the Mathematical Ninja: Converting radians to degrees (precisely)

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 = 270º1.

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 $\frac53 \pi$.2

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.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. If you prefer, you can do that as 540 / 2 = 270, but I prefer dividing first. Small numbers are easier to work with. []
  2. 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. []


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