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

76 thoughts on “Degrees vs. Radians

  1. The explanation put forward here is an excellent dissertation of the rationale for radians versus degrees. Consider trying to perform calculus, differential equations etc. when it is the angle that is unknown. The calculations would be quite cumbersome if the variable was in base 60 and the rest of the values were real numbers (base 10). By necessity the numbers would require base agreement to make any sense. In other words, all non angular values would need to be converted to base 60 or all angular values would need to be converted to base 10. By assuming radian measure we introduce a real valued variable (base 10) that is in base agreement with all other values in the expression (both numeric and angular as well as constant and variable). This coincides with the idea that radians are “dimensionless” or a “pure number” as previously explained. Essentially it is this base agreement that allows for the performance of the principles and formulas of calculus and differential equations to be performed as described in any number of textbooks, papers, etc.

  2. Excellent Article. You have made this read interesting and informative. Thank you. It paves a path to think in radians 🙂

  3. Radians are a form of tyranny laid upon us. Radians are irrational, both mathematically and otherwise. Your lies to attempt to deceive me into thinking in radians was logically flawed and tedious, you have irked me. Just like solving a equation with a matrix is idiotic, so is measuring angles in radians. One more think pi, as infatuated with it as you may be, is not at the “heart” of mathematics, only circles, and just as your arguments are is irrational.

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