Sal explains the definition and motivation for radians and the relationship between radians and degrees. Created by Sal Khan.
You are by now probably used to the idea of measuring angles in degrees. We use it in everyday language. We've done some examples on this playlist where if you had an angle like that, you might call that a 30-degree angle. If you have an angle like this, you could call that a 90-degree angle. And we'd often use this symbol, just like that. If you were to go 180 degrees, you'd essentially form a straight line. Let me make these proper angles. If you go 360 degrees, you have essentially done one full rotation. And if you watch figure skating on the Olympics, and someone does a rotation, they'll say, oh, they did a 360. Or especially in some skateboarding competitions, and things like that. But the one thing to realize-- and it might not be obvious right from the get-go-- but this whole notion of degrees, this is a human-constructed system. This is not the only way that you can measure angles. And if you think about it, you'll say, well, why do we call a full rotation 360 degrees? And there's some possible theories. And I encourage you to think about them. Why does 360 degrees show up in our culture as a full rotation? Well, there's a couple of theories there. One is ancient calendars. And even our calendar is close to this, but ancient calendars were based on 360 days in a year. Some ancient astronomers observed that things seemed to move 1/360 of the sky per day. Another theory is the ancient Babylonians liked equilateral triangles a lot. And they had a base 60 number system. So they had 60 symbols. We only have 10. We have a base 10. They had 60. So in our system, we like to divide things into 10. They probably liked to divide things into 60. So if you had a circle, and you divided it into 6 equilateral triangles, and each of those equilateral triangles you divided into 60 sections, because you have a base 60 number system, then you might end up with 360 degrees. What I want to think about in this video is an alternate way of measuring angles. And that alternate way-- even though it might not seem as intuitive to you from the get-go-- in some ways is much more mathematically pure than degrees. It's not based on these cultural artifacts of base 60 number systems or astronomical patterns. To some degree, an alien on another planet would not use degrees, especially if the degrees are motivated by these astronomical phenomena. But they might use what we're going to define as a radian. There's a certain degree of purity here-- radians. So let's just cut to the chase and define what a radian is. So let me draw a circle here, my best attempt at drawing a circle. Not bad. And let me draw the center of the circle. And now let me draw this radius. And let's say that this radius-- and you might already notice the word radius is very close to the word radians. And that's not a coincidence. So let's say that this circle has a radius of length r. Now let's construct an angle. I'll call that angle theta. So let's construct an angle theta. So let's call this angle right over here theta. And let's just say, for the sake of argument, that this angle is just the exact right measure so that if you look at the arc that subtends this angle-- and that seems like a very fancy word. But let me draw the angle. So if you were to draw the angle-- so if you look at the arc that subtends the angle, that's a fancy word. That's really just talking about the arc along the circle that intersects the two sides of the angles. So this arc right over here subtends the angle theta. So let me write that down. Subtends this arc, subtends angle theta. Let's say theta is the exact right size so that this arc is also the same length as the radius of the circle. So this arc is also of length r. So given that, if you were defining a new type of angle measurement, and you wanted to call it a radian, which is very close to a radius, how many radians would you define this angle to be? Well, the most obvious one, if you kind of view a radian as another way of saying radiuses, or I guess radii. Well, you say, look, this is subtended by an arc of one radius. So why don't we call this right over here one radian, which is exactly how a radian is defined. When you have a circle, and you have an angle of one radian, the arc that subtends it is exactly one radius long. Which you can imagine might be a little bit useful as we start to interpret more and more types of circles. When you give a degrees, you really have to do a little bit of math and think about the circumference and all of that to think about how many radiuses are subtending that angle. Here, the angle in radians tells you exactly the arc length that is subtending the angles. So let's do a couple of thought experiments here. So given that, what would be the angle in radians if we were to go-- so let me draw another circle here. So that's the center, and we'll start right over there. So what would happen if I had an angle-- if I wanted to measure in radians, what angle would this be in radians? And you could almost think of it as radiuses. So what would that angle be? Going one full revolution in degrees, that would be 360 degrees. Based on this definition, what would this be in radians? Well, let's think about the arc that subtends this angle. The arc that subtends this angle is the entire circumference of this circle. Well, what's the circumference of a circle in terms of radiuses? So if this has length r, if the radius is length r, what's the circumference of the circle in terms of r? Well, we know that. That's going to be 2 pi r. So going back to this angle, the length of the arc that subtends this angle is how many radiuses? Well, it's 2 pi radiuses. It's 2 pi times r. So this angle right over here, I'll call this a different-- well, let's call this angle x. x in this case is going to be 2 pi radians. And it is subtended by an arc length of 2 pi radiuses. If the radius was one unit, then this would be 2 pi times 1, 2 pi radiuses. So given that, let's start to think about how we can convert between radians and degrees, and vice versa. If I were to have-- and we can just follow up over here. If we do one full revolution-- that is, 2 pi radians-- how many degrees is this going to be equal to? Well, we already know this. A full revolution in degrees is 360 degrees. Well, I could either write out the word degrees, or I can use this little degree notation there. Actually, let me write out the word degrees. It might make things a little bit clearer that we're kind of using units in both cases. Now, if we wanted to simplify this a little bit, we could divide both sides by 2. In which case, on the left-hand side, we would get pi radians would be equal to how many degrees? Well, it would be equal to 180 degrees. And I could write it that way, or I could write it that way. And you see over here, this is 180 degrees. And you also see if you were to draw a circle around here, we've gone halfway around the circle. So the arc length, or the arc that subtends the angle, is half the circumference. Half the circumference is pi radiuses. So we call this pi radians. Pi radians is 180 degrees. And from this, we can come up with conversions. So one radian would be how many degrees? Well, to do that, we would just have to divide both sides by pi. And on the left-hand side, you'd be left with 1-- I'll just write it singular now. 1 radian is equal to-- I'm just dividing both sides. Let me make it clear what I'm doing here, just to show you this isn't some voodoo. So I'm just dividing both sides by pi here. On the left-hand side, you're left with 1. And on the right-hand side, you're left with 180/pi degrees. So 1 radian is equal to 180/pi degrees, which is starting to make it an interesting way to convert them. Let's think about it the other way. If I were to have 1 degree, how many radians is that? Well, let's start off with-- let me rewrite this thing over here. We said pi radians is equal to 180 degrees. So now we want to think about 1 degree. So let's solve for 1 degree. 1 degree, we can divide both sides by 180. We are left with pi/180 radians is equal to 1 degree. So pi/180 radians is equal to 1 degree. This might seem confusing and daunting. And it was for me the first time I was exposed to this, especially because we're not exposed to this in our everyday life. But what we're going to see over the next few examples is that as long as we keep in mind this whole idea that 2 pi radians is equal to 360 degrees, or that pi radians is equal to 180 degrees, which is the two things that I do keep in my mind. We can always re-derive these two things. You might say, hey, how do I remember if it's pi/180 or 180/pi to convert the two things? Well, just remember, which is hopefully intuitive, that 2 pi radians is equal to 360 degrees. And we'll work through a bunch of examples in the next video, to just make sure that we're used to converting one way or the other.