Why angles are measured in degrees

Really, anything that measures the openness. So when you go into trigonometry, you'll learn that you can measure angles, not only in degrees, but also using something called "radians. So let's measure this other angle, angle BAC.

So once again, I'll put A at the center, and then AC I'll put along the 0 degree edge of this half-circle or of this protractor.

And then I'll point AB in the-- well, assuming that I'm drawing it exactly the way that it's over there. Normally, instead of moving the angle, you could actually move the protractor to the angle. So it looks something like that, and you could see that it's pointing to right about the 30 degree mark. So we could say that the measure of angle BAC is equal to 30 degrees.

And so you can look just straight up from evaluating these numbers that 77 degrees is clearly larger than 30 degrees, and so it is a larger angle, which makes sense because it is a more open angle. And in general, there's a couple of interesting angles to think about. If you have a 0 degree angle, you actually have something that's just a closed angled. It really is just a ray at that point.

As you get larger and larger or as you get more and more open, you eventually get to a point where one of the rays is completely straight up and down while the other one is left to right. So you could imagine an angle that looks like this where one ray goes straight up down like that and the other ray goes straight right and left.

Or you could imagine something like an angle that looks like this where, at least, the way you're looking at it, one doesn't look straight up down or one does it look straight right left.

But if you rotate it, it would look just like this thing right over here where one is going straight up and down and one is going straight right and left. And you can see from our measure right over here that that gives us a 90 degree angle.

It's a very interesting angle. It shows up many, many times in geometry and trigonometry, and there's a special word for a 90 degree angle. It is called a "right angle. We would call this a "right angle. You draw a little part of a box right over there, and that tells us that this is, if you were to rotate it, exactly up and down while this is going exactly right and left, if you were to rotate it properly, or vice versa.

And then, as you go even wider, you get wider and wider and wider and wider until you get all the way to an angle that looks like this. So you could imagine an angle where the two rays in that angle form a line. So let's say this is point X. This is point Y, and this is point Z. You could call this angle ZXY, but it's really so open that it's formed an actual line here.

Z, X, and Y are collinear. This is a degree angle where we see the measure of angle ZXY is degrees. And you can actually go beyond that. So if you were to go all the way around the circle so that you would get back to degrees and then you could keep going around and around and around, and you'll start to see a lot more of that when you enter a trigonometry class. Now, there's two last things that I want to introduce in this video.

There are special words, and I'll talk about more types of angles in the next video. But if an angle is less than 90 degrees, so, for example, both of these angles that we started our discussion with are less than 90 degrees, we call them "acute angles.

So that is an acute angle, and that is an acute angle right over here. They are less than 90 degrees. What does a non-acute angle look like? And there's a word for it other than non-acute. Well, it would be more than 90 degrees. So, for example-- let me do this in a color I haven't used-- an angle that looks like this, and let me draw it a little bit better than that. An angle it looks like this. So that's one side of the angle or one of the rays and then I'll put the other one on the baseline right over here.

Clearly, this is larger than 90 degrees. If I were to approximate, let's see, that's , , , almost So let's call that maybe a degree angle.

We call this an "obtuse angle. It's nice and small. In trigonometry, radians are used most often, but it is important to be able to convert between any of the three units. A revolution is the measure of an angle formed when the initial side rotates all the way around its vertex until it reaches its initial position. Thus, the terminal side is in the same exact position as the initial side. In trigonometry, angles can have a measure of many revolutions--there is no limit to the magnitude of a given angle.

A revolution can be abbreviated "rev". A more common way to measure angles is in degrees. There are degrees in one revolution. Degrees can be subdivided, too. One degree is equal to 60 minutes, and one minute is equal to 60 seconds. Therefore, an angle whose measure is one second has a measure of degrees. When perpendicularity is discussed, it is most often defined as a situation in which a 90 degree angle exists. Often degrees are used to describe certain triangles, like and triangles. As previously mentioned, however, in most cases that concern trigonometry, radians are the most useful and manageable unit of measure.

Degrees are symbolized with a small superscript circle after the number measure. A radian is not a unit of measure that is arbitrarily defined, like a degree.