Degrees vs. Radians

An Open Letter to Students Wondering, “Why Do We Use Radians Instead of Degrees?”

Dear Student,

Let me start with a question of my own. Did you know that you speak Babylonian?

You may not think that you do – you can’t ask for directions to a Babylonian bathroom, or order lunch off of a Babylonian menu, or ask a good-looking Babylonian out for coffee. But you’ve inherited Babylon’s legacy nevertheless.

You’ve probably noticed that our systems of counting and measuring are largely base 10. There are 10 years in a decade, 10 decades in a century, and 10 centuries in a millennium. But take a look at shorter time lengths: We don’t carve up the day into 10 hours, each 100 minutes long, even though this would make perfect sense. Instead, we divide our time up by multiples of 60 – 60 seconds in a minute, 60 minutes in an hour.

Why do we do this? Because our Babylonian uncles used a base 60 number system, and we’ve been following their example ever since.

Babylon left another relic in our system of measurements, one that requires a little deeper explanation, and which lies at the heart of your question (yes, I’m getting there): the degree.

***

Angles are everywhere. They form when light bounces off of a mirror, when a bird dives towards the water, when a lamp casts a shadow on a distant wall. And mathematicians, being the go-getters that they are, need a way to measure angles.

It’s clear from looking at two angles which one is bigger:

But without any numbers, it’s hard to say how much bigger. That’s where angle measurements come in.

We can think of any angle as a piece of a circle. For example, a right angle (like the corner of a rectangle) is one quarter of a circle, and the angle in an equilateral triangle takes up one sixth of a circle:

So, how do we measure angles? Well, the Babylonians had an idea. They decided to cut a circle into 360 pieces, and call one of those pieces a degree.

This makes it easy to talk about the size of an angle. Since there are 360 degrees in a whole circle, a right angle will be one-quarter as many degrees – i.e., 90o.

Most people learn this system so well, and at such a young age, that it becomes second nature. Skateboarders pull 540o’s, a businessman looking to change plans will call for a 180o, and at one point in our political past, Americans chanted “Fifty-four forty or fight!” to demand that our border with Canada be set at 54o40’ of latitude (where 1’ = 1/60th of a degree).

What this familiarity disguises is that the number 360 is totally arbitrary, chosen simply because the Babylonians preferred multiples of 60. Why not divide the circle into 100 pieces, or 5 pieces, or 400 pieces? (In fact, the 400-piece system does exist – 1/400th of a circle is called a gradian, and it’s that weird mode on your calculator that you never use.) There’s no mathematical reason to pick 360, or any specific number, for that matter. It’s fundamentally just a matter of taste.

***

As it turns out, there’s a better way. But to explore it, we’ve got to lift ourselves out of this rut of thinking. No more dividing the circle into some arbitrary number of units.

Instead, let’s draw a whole circle, and throw in a radius for good measure.

Now, let’s take that radius and wrap it around the outside of the circle. See how it forms an angle? We’ll call that angle 1 radian.

Here’s an angle that’s 2 radians, for comparison.

And here’s an angle that’s 3 radians. Notice that it’s almost half a circle (what we used to call 180o), but not quite.

This raises a question: how many radians are there in a circle? Draw it out, and you’ll find that it’s not a nice, round integer:

Looks like a little more than 6. But how much more?

For that, we’ll need to dredge up a geometric formula relating the circumference and the radius:

C = 2πr

In other words, the circumference is equal to the length of the radius times 2π (where π is roughly 3.14. For a better – but still imperfect – approximation, try this).

This tells us exactly how many radians there are in a circle: 2π!

Knowing this, we can now convert between radians and degrees – just as we can convert between miles and kilometers, or Fahrenheit and Celsius. Radians become a perfectly valid, usable measure of angles.

***

But I know you’re not satisfied with that. You’re sharp-witted and wary of being made to learn new things. You want to know: What was wrong with degrees? Arbitrary or not, they let us work with integers ( like 30o) instead of nasty ratios involving irrational numbers (like π/6). Degrees are warm, friendly, familiar. Why ditch them in favor of this bizarre radian?

Here’s the best answer I can give you: Degrees are fine for everyday measurements. But Trigonometry marks a turning point in math, when the student lifts his gaze from the everyday towards larger, more distant ideas. You begin exploring basic relationships, deep symmetries, the kinds of patterns that make the universe tick. And to navigate that terrain, you need a notion of angles that’s more natural, more fundamental, than slicing up the circle into an arbitrary number of pieces. The number π, strange though it may seem, lies at the heart of mathematics. The number 360 doesn’t. Clinging to that Babylonian artifact will only distract you and obscure the elegant truths you’re searching for.

 ***

Just like you, I learned to speak Babylonian long before I encountered radians. And for years, Babylonian remained my native tongue – to give an angle in radians first required an act of mental translation. So if you’re resistant towards radians, nostalgic for Babylon, I can sympathize.

Still, when push comes to shove, radians can take you places that degrees simply can’t. That’s why, when my Trig students give an angle in degrees instead of radians, I tell them: “I’m sorry, I don’t speak Babylonian.”

I hope this helps. When in doubt, remember: 180= π radians.

Best,
Ben

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14 thoughts on “Degrees vs. Radians

    • Thanks! My hazy memory is that radians can be treated as unit-less in a lot of contexts… although I can’t remember why at the moment.

      EDIT: You’re right that the “rad” is necessary. Radians are unitless because they’re arc length divided by radius length (so the units of length cancel). But the same is true of degrees (which are arc length divided by 1/360th of circumference). So including the units is the right move. Will edit accordingly.

  1. Angle is the ratio of length of an arc in a circle and of rsdiud its radius., Both quantities have the dimension of Length. 1 Radian subtends arc of length of the radius.

  2. It’s probably worth noting that our year has 365 days, but the Babylonian year had 365 days (split into 12 months of 30 days each plus 5 extra days for holidays) which may have influenced why they chose 360 as the number of degrees in a circle. 360 is also convenient because it divides so many ways into nice whole unit portions.

    • Yeah, the many factors of 360 do make it nice to use. I wonder if that’s part of the reason gradians never caught on – 400 has only two prime factors, whereas 360 has three.

      The Babylonian version of the year sounds kinda nice – I’d love to have equal-length months, with five bonus days.

      • I want to say that a former Roman calendar had something like that: 12 months of 30 days, then a week-long festival to get back on track (on that note why ISN’T Leap Day a Federal holiday?). Or maybe that was Egyptian. The question then becomes…why did anyone ever pick SEVEN as the divisor for weeks?

        With no evidence whatsoever, I’d guess the thought behind the 1/400th has to do with the historical definition of a meter. Originally defined by a study from the French Academy of Science in the 18th century as a universally human unit to be embraced by all countries of the world:
        –1 meter = 1 ten millionth of a quarter meridian
        –1 meter = 1/40,000,000 of the (meridianal) circumference of the earth
        –100 km = 1/400th of the circumference of the earth.
        So now you can sail your ships (or fight the Canadians) over lovely gradian arc lengths of 100 km, and each lon-lat grid also has an area of 1 hectare (ignoring curvature).

        Of course, although a solid effort for the 1790’s, the measurements weren’t quite perfect (the meter was never redefined, so modern measurement places the Earth’s meridianal circumference at 40,008 km). And, more to the point, the Earth isn’t perfectly spherical (equatorial circumference is 40,075 km).

        • Ah, cool. I’d never thought about the gradian as corresponding to nice arc lengths going around the world.

          There’s also a quicker (though maybe no more satisfying) explanation for the 400: it makes a right angle 100 gradians, which feels suitably metric.

          As for the seven-day week, Wikipedia is as stumped as the rest of us. It’s a quarter of a lunar cycle, which I guess is nice if you’re on a lunar calendar, but Wikipedia pokes some holes in that theory, too.

          http://en.wikipedia.org/wiki/Seven-day_week

  3. One effective interactive way to introduce radian measure is to use your body to measure the angle subtended by a distant object. To make the measurement, extend your arm and sight along your thumb. Of course, our arms and thumbs are different sizes, so to standardize the measurement, students can be prodded to realize that they need to measure the ratio of the height of the object (along the thumb) to the distance from the eye to the thumb. This ratio gives the angle in radians. For instance, if my thumb is 60 cm from my eye and my thumbnail is 2 cm long, an object that appears the same size as my thumbnail subtends an angle of 2 cm / 60 cm = 1/30 radian. Though I could convert that result to degrees, it’s more useful in radians. If the object I’m measuring is about a football field’s length away, I can find its height directly from my angle measurement: 1/30 x 100m, or about 3.3 m high. Presented and used this way, radians are natural, useful, and memorable, making it interesting for students to solve the challenging question of how these natural angle units relate to angle measurements in degrees.

    • I like that. Radians are built around arc length – a quantity we rarely use in trig classes after week 1. But this is a nice way to make finding arc length, as you say, “natural, useful, and memorable.”

  4. It’s so hard to think in terms of degrees now that I’ve been in higher-level math courses for the last few years! All those physicists out there need to standardize to radians. (OK, I’m just kidding.)

    Another good grump is the whole pi vs. tau issue: Other than the formula for area of a circle, why not use tau instead of pi? It would be a lot simpler… :)

    • Yeah, I like tau. It would make the area formula less elegant, but it’d make the formula for circumference more elegant (tau * r), among the other improvements.

  5. When I was a trigonometry student, I was convinced that we should all ditch radians AND degrees, and just use the “turn” as our unit of angle. A quarter of a turn, a half of a turn, 3/7 of a turn – what could be easier?

    Only recently did I realize what you pointed out in one of the drawings here: if you were using any units other than radians, cos(x) would not be the derivative of sin(x). If x were in “turns,” the derivative of sin(x) would be 2*pi*cos(x). The solutions to f” = -f, the functions that you and I know simply as sin(x) and cos(x), would be called cos(x/(2pi)) and sin(x/(2pi)). And Euler’s Formula would be e^(i*2*pi*x) = cos(x) + i*sin(x).

    …Actually, Euler’s Formula doesn’t look too bad in its new form. Especially if you’re using tau: e^(i*tau*x) = cos(x) + i*sin(x). The polar form of a complex number would be re^(i*tau*theta). It seems to me like all the math we do now would still be doable with “turns” instead of radians. Just a little bit uglier.

    I wouldn’t be surprised if there was another planet in our galaxy full of sentient beings who adopted turns instead of radians as the conventional angle unit of higher math, because it seemed like the obvious thing to do at the time. Now, centuries later, one of their mathematicians has written up a Radian Manifesto explaining why everything would be so much prettier if they would just start using this weird alternate unit of angle instead of the turn. But of course it’s far too late for them to change: everyone uses turns as the assumed unit for trig functions in all mathematical publications, and every calculus student has memorized that the derivative of sin(x) is tau*cos(x), and how on earth could they make the transition now? We humans should pat ourselves on the back for not making that mistake.

  6. Pingback: thetawelle » Blog Archiv » Einheitskreis, Bogenmaß, Polarkoordinaten - frontier of science, technology & future

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