Angle measurement & circle arcs

We already know that an angle is formed when two rays share a common endpoint. So, for example, let's say that this is one ray right over here, and then this is one another ray right over here, and then they would form an angle. And at this point right over here, their common endpoint is called the vertex of that angle. Now, we also know that not all angles seem the same. For example, this is one angle here, and then we could have another angle that looks something like this. And viewed this way, it looks like this one is much more open. So I'll say more open. And this one right over here seems less open. So to avoid having to just say, oh, more open and less open and actually becoming a little bit more exact about it, we'd actually want to measure how open an angle is, or we'd want to have a measure of the angle. Now, the most typical way that angles are measured, there's actually two major ways of that they're measured. The most typical unit is in degrees, but later on in high school, you'll also see the unit of radians being used, especially when you learn trigonometry. But the degrees convention really comes from a circle. So let's draw ourselves a circle right over here, so that's a circle. And the convention is that-- when I say convention, it's just kind of what everyone has been doing. The convention is that you have 360 degrees in a circle. So let me explain that. So if that's the center of the circle, and if we make this ray our starting point or one side of our angle, if you go all the way around the circle, that represents 360 degrees. And the notation is 360, and then this little superscript circle represents degrees. This could be read as 360 degrees. Now, you might be saying, where did this 360 number come from? And no one knows for sure, but there's hints in history, and there's hints in just the way that the universe works, or at least the Earth's rotation around the sun. You might recognize or you might already realize that there are 365 days in a non-leap year, 366 in a leap year. And so you can imagine ancient astronomers might have said, well, you know, that's pretty close to 360. And in fact, several ancient calendars, including the Persians and the Mayans, had 360 days in their year. And 360 is also a much neater number than 365. It has many, many more factors. It's another way of saying it's divisible by a bunch of things. But anyway, this has just been the convention, once again, what history has handed us, that a circle is viewed to have 360 degrees. And so one way we could measure an angle is you could put one of the rays of an angle right over here at this part of the circle, and then the other ray of the angle will look something like this. And then the fraction of the circle circumference that is intersected by these two rays, the measure of this angle would be that fraction of degrees. So, for example, let's say that this length right over here is 1/6 of the circle's circumference. So it's 1/6 of the way around the circle. Then this angle right over here is going to be 1/6 of 360 degrees. So in this case, this would be 60 degrees. I could do another example. So let's say I had a circle like this, and I'll draw an angle. I'll put the vertex at the center of the angle. I'll put one of the rays right over here. You could consider that to be 0 degrees. Or if the other ray was also here, it would be 0 degrees. And then I'll make the other ray of this angle, let's say it went straight up. Let's say it went straight up like this. Well, in this situation, the arc that connects these two endpoints just like this, this represents 1/4 of the circumference of the circle. This is, right over here, 1/4 of the circumference. So this angle right over here is going to be 1/4 of 360 degrees. 360 degrees divided by 4 is going to be 90 degrees. At an angle like this, one where one ray is straight up and down and the other one goes to the right/left direction, we would say these two rays are perpendicular, or we would call this a right angle. And the way that we oftentimes will denote that is by a symbol like this. But this literally means a 90-degree angle. Let's do one more example. Let's do one more example of this, just to make sure that we understand what's going on. Actually, at least one more example. Maybe one more if we have time. So let's say that we have an angle that looks like this. Once more, I'm going to put its vertex at the center of the circle. That's one ray of the angle. And let's say that this is the other ray. This right over here is the other ray of the angle. I encourage you to pause this video and try to figure out what the measure of this angle right over here is. Well, let's think about where the rays intersect the circle. They intersect there and there. The arc that connects them on the circle is that arc right over there. That is literally half of the circumference of the circle. That is half of the circumference, half of the way around of the circle, circumference of the circle. So this angle is going to be half of 360 degrees. And half of 360 is 180 degrees. And when you view it this way, these two rays share a common endpoint. And together, they're really forming a line here. And let's just do one more example, because I said I would. Let me paste another circle. Let me draw another angle. Let me draw another angle. So let's say that's one ray of the angle, and this is the other ray. This is the other ray of the angle right over here. And we care. There's actually two angles that are formed. There's actually two angles formed in all of these. There's one angle that's formed right over here, and you might recognize that to be a 90-degree angle. But what we really care about in this example is this angle right over here. So once again, where does it intersect the circle? We care about this arc right over here, because that's the arc that corresponds to this angle right over here. And it looks like we've gone 3/4 around the circle. So this angle is going to be 3/4 of 360 degrees. 1/4 of 360 degrees is 90, so three of those is going to be 270 degrees.