Why radians rock (and degrees don’t)

If I could wave a magic wand and overhaul just one thing to make the world a better place, I’d have a tough choice. Would I get rid of the QWERTY keyboard in favour of a more sensible layout? Would I make the English language fonetik? Would I take maths notation and make it a bit more obvious? Would I decimalise time? Tempting and worthwhile as these suggestions are, I would use my wand more wisely: I would abolish degrees for measuring angles and insist that everyone use radians.

Not just because degrees are nothing to do with temperature degrees and having one word for two different things is silly, nor because they’re nothing to do with university degrees and having one word for three things is even sillier, but because 360 has nothing to do with circles.

Idiotic Sumerians

Honestly, what were the Sumerians thinking? One degree per day of the year? For one, what does that have to do with anything, and for two, could they not even get THAT right to within one per cent? Or because it divides nicely by lots of numbers? Tell that to the septagon – and for that matter, all manner of nicely-divisible imperial measurements that are dying the same kind of embarrassing death the degree should also suffer.

Instead, grown-ups get to use radians, which are superior to the degrees in every possible way. Like QWERTY vs Dvorak and Windows vs Mac, the only reason anyone would prefer degrees is inexplicable popularity.

$\pi$ in the sky

Since $\pi$ crops up so often in circle formulas, don’t you think it makes sense to measure angles using something $\pi$-related? A whole circle is $2\pi$ radians – same as the circumference if the radius is one unit. A semi-circle is $\pi$ radians. Simple!

You often see radian angles given as fractions of $\pi$. Don’t be a baby, that’s just telling you how much of a semi-circle you have. $\frac{\pi}{3}$ is a third of a semi-circle. $\frac{\pi}{2}$? Half a semi-circle, or a right angle.

Not convinced? Let’s compare some formulas. Take a sector of a circle – let’s say 70º in the middle. How long is the curvy bit? In degrees, you say “well, that’s $\frac{70}{360}$ of a circle, and the whole thing is $2\pi r$, then I have to multiply that by the fraction and my eyes have gone all wibbly.” In radians, you say “1.2 radians… multiply it by the radius… job done. Shall I get you an aspirin?”

Oh, you wanted the area instead? Well, try it in degrees on your calculator. Meanwhile, I’ll do $1.2 \frac{r^2}{ 2}$ and grab a coffee while I wait for you to finish.

I dare you to differentiate in degrees

But wait – there’s more! For small angles, $\sin(x)$ is roughly the same as $x$ – but only if you do it in radians. Try it – put your calculator in radian mode and type $\sin(0.01)$. It’s 0.0099 or something, isn’t it? Oh, my mistake. 0.0099998. The approximate formula is off by about $\frac{1}{500}$ of a percent. What’s $\sin(0.01^\circ)$? I have no idea, and neither do you.

“Big deal,” I can hear you shrug, “so you can estimate some things.” Well, there are some pretty deep consequences of that – such as being able to do calculus. The gradient of $\sin(x)$ is $\cos(x)$ – if you do it in radians. In degrees, you end up with crazy multipliers all over the place – all the $\pi$s you were probably trying to avoid by using degrees in the first place.

So, let’s recap: radians are based on the actual size of a circle, make all the formulas easy and make it possible to do calculus, and are basically responsible for the last 300 years of scientific progress. Meanwhile, degrees… are based on a 4,000 year-old calculation of the length of the year.

Which they got wrong.

Colin

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.

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