I’ll hold up my hands here and say: most of the secrets of the mathematical ninja are guides to showing off. Few of them have much practical use beyond making everyone else in the class look at you funny. This one - converting degrees into radians - is different.

If you’ve read my article on why radians rock, you’ll know exactly where my loyalties in the radians-versus-degrees debate lie - and unlike with logs base ten, there’s no way back for degrees. From C2 onwards, you need to be able to use radians with comfort and panache, if only because calculus doesn’t work in degrees.

And every year, my C2 students ask me ‘how do you convert radians to degrees, again?’ and I say ‘you don’t,’ because there’s almost never a reason to. Converting degrees into radians, on the other hand, is a different story.

One radian is about the same as 57.3 degrees (more precisely, it’s $\frac{180}{\pi}$). The showing-off ninja might say “Aha! $\frac{180}{\pi}$ is close to $180 \div \frac{22}{7}$, which is $\frac{1260}{22}$, or $\frac{630}{11}$ (= 57.27). That’s actually a really good approximation, if you can pull it off (it’s about 0.05% low) - but even a real-life mathematical ninja would find the sums a bit tricky on that one.

Instead, I tend to go for a radians-into-degrees factor of $\frac{400}{7}$ (= 57.14), which is a little less accurate (-0.3%), but uses simple numbers. To get a rough radians value from a number of degrees, do this:

1. Divide by 4
2. Multiply by 7
3. Divide by 100 by moving the decimal point left by two places.

So, to turn 115 degrees into radians, you divide by 4 (29 less a quarter), multiply by 7 (203 less 7/4, 201.25) and put the point in - 2.0125. The actual answer is 2.0071, so we’re right to 3 sig fig.

If you absolutely insist on going backwards, you just do the same thing in reverse: multiply by 100, divide by 7 and multiply by 4 (in any order you like). That means 1.2 radians is about the same as $\frac{480}{7} = 68.6 ^\circ$. The right answer is 68.755. Off by two in the third sig fig; again, that’s an error of about 0.3%

So, why is this useful? It’s a good check to see whether you’ve ‘gone the right way’ in your conversion, especially if you’re not prepared to use some common sense to decide whether 5,000 radians is a reasonable angle. And converting degrees into radians is a great way to impress members of the same/opposite sex, as you prefer.