What Are Radians and Degrees?
What’s the difference between radians and degrees? Where did these units for measuring angles come from? And how do you convert from one to the other? Keep on reading to find out!
Jason Marshall, PhD
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What Are Radians and Degrees?
Now that we’ve learned the basics of sines, cosines, and tangents, we’re ready to dig deeper into the world of trigonometry. But before we dig too far, we need to take a little time to understand how angles—which are a key idea in trigonometry—are quantified and measured.
So prepare yourself for some excitement because today we’re diving into the world of radians and degrees. We’ll talk about where these units come from, how to convert between them, and where you’re most likely to see each..
What Are Radians?
Meauring angles in radians is one of those things in math that seems kind of strange and maybe even a little stupid…until you realize that it actually makes perfect sense. Of course, most of us never realize that it makes perfect sense because we never learn to fully appreciate the simple logic behind the radian. So let’s remedy that.
To understand radians, let’s think about drawing a circle. You can draw a circle by attaching one end of a piece of string to a pencil, pinning the other end down to a sheet of paper, and then dragging the taut string around. As it turns out, this string isn’t just a string—it’s length is the same as the radius of the circle. And if you take that string and place it along the circle’s circumference, you’ll find that it takes a little more than 3.14—aka, π—of these radii to go half-way around.
Radians give a very simple way to think about angles in terms of circles.
If you now lay that radius string along some portion of your circle and draw a pair of lines from the center of the circle to the ends of the string, the angle between those lines will be 1 radian. A string that’s half as long gives a 0.5 radian angle, a string that’s 20% as long gives a 0.2 radian angle, and a string that’s approximately 3.14 times as long gives an angle spanning half the circle—which is about π radians.
As you can see, radians give a very simple way to think about angles in terms of circles. Which makes them not nearly as crazy as you thought, right?
What Are Degrees?
Although most of us are more comfortable thinking in terms of degrees—which means we’re inclined to think of degrees as being less “weird” than radians—degrees actually are the more bizarre way to measure angles. After all, we’ve seen that the 2π radians of the angle formed around a complete circle have a mathematically sensible origin—something that cannot be said about the 360 degrees around a circle.
Which leads us to a very simple—but also very good—question: Why are there 360 degrees in a circle? And the answer is: We’re not exactly sure, but it most likely has to do with the same history that has given us 24 hour days, 60 minute hours, and 60 second minutes.
Degrees are only “natural” to us because we’re used to them.
Namely, the fact that the ancient Babylonians used a base-60 number system ends up meaning that these are the sorts of numbers you’d expect them to have come up with. And they did! This, coupled with the fact that a year on Earth is pretty close to 360 days—a time over which ancient folks no doubt watched stars travel in their annual circular patterns—means that we ended up with 360 degrees in a circle.
Which goes to show that degrees are only “natural” to us because we’re used to them. And that they really are a rather strange—being almost accidental—way for us to keep track of angles!
How to Convert Between Radians and Degrees
While some of you may prefer to work in terms of radians and others may prefer to work in terms of degrees, the truth is that you won’t always be given angles in your preferred units. Which means that you may occasionally need to convert from one to the other.
Fortunately, that’s not too tough. In fact, all you really need to know is that the angle formed by travelling half-way around a circle is either 180 degrees or π radians. In other words,
π radians = 180 degrees
If we divide both sides of this equation by the approximate value of π, this says that 1 radian is approximately equal to 57.3 degrees. Following the logic we learned in the episode on converting units, to convert from:
- degrees to radians: multiply by π and divide by 180
- radians to degrees: multiply by 180 and divide by π
That’s all there is to it! Some might say it’s as easy as π (yes, I went there).
Should You Use Radians Or Degrees?
Since we now know how to convert between radians and degrees, we’re completely free to use either whenever we’d like. But, you might be wondering, which should you usually use? Well, it depends—sometimes radians are more convenient than degrees, and sometimes it’s the other way around. And sometimes you don’t have any choice.
Let’s start with degrees. For most of our everyday dealings with angles, degrees is the hands-down winner. After all, natural or not, most of us have a good intuitive grasp of what 45, 90, 120, and 180 degree angles look like. And if you’re doing something requiring a protractor (which, I must sadly say, I don’t think I’ve used since elementary school), you’re going to be working in degrees.
On the other hand, if you’re working with angles that are being plugged into sine, cosine, or tangent trigonometric functions on a calculator, you must work in terms of radians since that’s how those functions are defined. Finding the sine or cosine of an angle written in degrees will absolutely 100% give you a nonsensical (and very wrong) answer—so don’t try to do it!
Otherwise, you’re free to use whatever unit your heart desires.
Wrap Up
Wrap UpOkay, that’s all the math we have time for today.
Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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Protractor and 360 degrees images courtesy of Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.