Math Antics - Angles & Degrees

Hello and welcome to Math Antics. In our last geometry video, we learned some important things about

angles

. One of the things we learned was that

angles

come in different sizes. Some are big and some are small. Well, in this video we will learn how we can tell exactly how big or small an angle is. Let's learn how angles are measured. You probably already know a lot about measurements, such as knowing how to measure the length of something with a ruler or tape measure. And the units you would use would be inches, centimeters or something like that, right?

But when it comes to angles, we can't use a ruler to measure them, nor use units like centimeters. And that's because angles don't have to do with length, but with rotation. And to measure how much something spins, we use a special unit called

degrees

. Wait a second, I thought

degrees

were used to measure how hot or cold something is. You know, like, "it's 100 degrees outside today!" Ah, that's a good point, smart-looking friend. The word title is actually used for a lot of different things, so it can be a little confusing at times. It makes more sense if you think of a title as a small amount of something.

For temperature, a degree is a small amount of heat. But for angles, a degree is a small amount of rotation. And there is a special

math

ematical symbol for degrees that we can use instead of writing the word "degrees" over and over again. It's this little circle that you put after the number and near the top. To see how we use degrees to measure angles, let's get two rays pointing in exactly the same direction. Next, let's place one ray directly on top of the other, so it looks like there's only one ray there, even though there are actually two.

Now, let's take the beam above and rotate it a little counterclockwise. This point on the ray will be our axis (or center) of rotation. It is like the dot in the center of a clock that remains stationary while the hands rotate around it. Our rays now make an angle measuring 1 degree, and as you can see, 1 degree is a really small angle. We need to get closer to see that it really is an angle. In fact, you might wonder if there could be any angle smaller than 1 degree. Yes, there surely are. And we saw one just a second ago.

Before we rotated our top ray, when our rays were exactly on top of each other, that was a zero degree angle. And there are a whole range of small fractional angles between 0 and 1 degree, but we're not going to learn about them in this video. Instead, we're going to continue rotating our top ray and watch the angle get larger and larger. This special reading here will tell us how many degrees our angle measures. Now let's start slowly. 1 grade, 2, 3, 4, 5, 6, 7, 8, 9 and 10, now let's hold it there for a second. This is what 10 degrees looks like. 10 grades?! That's freezing! Hey. I guess you're not as smart as I thought after all.

So we can see that an angle of 10 degrees is still a very small angle. So let's move on, but this time a little faster. That's 15 degrees, 20, 25, 30, 35, 40 and 45. Now, 45 degrees is a special angle because it is exactly half of a right angle. If we draw a right angle in the same place, you can see that our ray cuts it into two equal parts. So if 45 is half a right angle, can you guess how many degrees a right angle is? Let's keep rotating to see if you're right. 50, 60, 70, 80, and 90. Yes, a right angle measures exactly 90 degrees, and it's very important to memorize that because right angles are used all the time in geometry.

Well, do you remember from our last video that all angles less than a right angle are called acute angles? That means that all the angles we've seen so far that are between 0 and 90 degrees (such as 10, 30, 45, 60, and so on) are acute angles. But, if we continue rotating our ray more than 90 degrees, we will begin to form obtuse angles, because they are larger than a right angle. Ready? Here we go. 100 degrees, 110, 120, 130, 140, 150, 160, 170 and 180. Ah ha, sound familiar? Yes. It is a right angle, as we learned in the last video. The rays point in exactly opposite directions and the angle they form is 180 degrees.

And that is also a very important angle measurement that must be memorized. Now, before we continue, let's quickly review the important angles and regions we have discussed so far. Our angle measurement is zero degrees when the rays point in the same direction. They are 90 degrees when they are perpendicular and form a right angle. And they are 180 degrees when they point in opposite directions and form a right angle. In this region (between 90 and 180) we find obtuse angles. And in this region (between 0 and 90) we find acute angles. An important acute angle is 45 degrees, since it is half of a right angle.

Okay, so let's continue turning more than 180 degrees. Our angle reading continues to increase and the next major angle we reach is this one, 270 degrees. It also forms a right angle, but points down instead of up. Let's move on because another really important angle is just around the corner. And it's about to appear right now! We have rotated our ray completely around the axis and now it is back to the starting point. Now you may be wondering, "If we go back to square one, why does our counter read 360 degrees instead of 0 degrees like before?" The answer is that even though our beams return to the same place, we had to rotate our top beam 360 degrees to get there.

And you can see that our angular arc now forms a complete circle. So 360 degrees is the angle that represents a complete circle! By turning 360 degrees, you will go all the way around the circle to the point where you started. Okay, now that you know what degrees are and you've seen how they relate to the size of an angle, we need to learn how to measure an angle without this fancy reading that we have here. Just as you can use a ruler to measure the length of a line, a special tool called a protractor can be used to measure angles.

A protractor is similar to a ruler, but it is curved into a semicircle so that it can measure rotation around a point on the axis. A protractor also has a ruler with a hole or point in the middle that represents the axis or center of rotation. So if you're given a mystery angle (like this one) and you want to measure how many degrees it is, simply place the protractor on top so that the axis point lines up with the intersection of your rays, like this. Then you make sure that one of the spokes lines up with the straight line of the protractor.

And lastly, you look to see where the other ray crosses the curved part and read what angle measure it lines up with. As you can see, this angle is 50 degrees. Alright, there's one more thing I want to teach you in this video because you'll probably see this type of geometry problem in your homework or exams. Do you remember what the complementary and supplementary angles are from the last video? Complementary angles combine to form a right angle and supplementary angles combine to form a straight angle. Well, now that we know that a right angle measures 90 degrees and a right angle measures 180 degrees, we can use that information to solve problems that have unknown angles, like this one.

Shows two angles (A and B) that combine to form a right angle. The problem tells us that angle A measures 30 degrees and wants us to find out what angle B is. Fortunately, it is now easy to solve because we know that a right angle measures 90 degrees, so we know what the total of both angles must be. That means that to find angle B, all we do is take the total (which is 90 degrees) and subtract angle A (which is 30 degrees) and whatever is left will be angle B. So, 90 - 30 = 60 .Angle B is 60 degrees. Now let's try this problem.

Use the same idea, but this time with a right angle. The straight angle is divided into two smaller angles. (again, angle A and angle B) And again, the problem tells us that angle A is 70 degrees and wants us to figure out what angle B is. Well, we know that the total of both angles must be 180 degrees because we just learned that that's how big a right angle is. So if we take that total (180 degrees) and subtract angle A (which is 70 degrees), whatever is left after subtracting must be angle B. So 180 - 70 = 110. Cool, right? And now you can see why it is important to know how degrees work in geometry.

They can tell us how big the angles are or how much something is rotated. Well, that's all I have for you in this video. But don't worry, there's a lot more geometry where that came from. So I will continue with my next video and you will continue practicing what you have learned. Thanks for watching Math Antics and see you next time. Learn more at

math

ematics.com.