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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!

By
Jason Marshall, PhD,
January 25, 2014
Episode #183

Page 1 of 2

ProtractorNow 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.

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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?

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