Measuring angles in degrees How to measure an angle in degrees
Measuring angles in degrees
- Now that we know what an angle is, let's think about how we can measure them--and we already
- hinted at one way to think about the measure of an angle in the last video when we said:
- "Look, this angle XYZ seems more open than angle BAC so maybe the measure of angle XYZ
- should be larger than the angle of BAC", and that is exactly the way we think about the measures of angles,
- but what I want to do in this video is come up with an exact way to measure an angle.
- So what I've drawn over here is a little bit of a half circle, and it looks very similar
- to a tool that you can buy at your local school supply store to measure angles.
- So this is actually a little bit of a drawing of a protractor, and what we do with something like a protractor
- --and you can even construct one with a piece of paper--is we've taken a half circle right here,
- and we've divided it into 180 sections, and each of these marks marks ten of those sections,
- and what you do for any given angle is you put one of the sides of the angle--so each of the rays of
- the angle are considered one of its sides--so you put the vertex of the angle at the center of
- this half circle, or if you're dealing with an actual protractor at the center of that protractor
- and then you put one side along the zero mark. So I'm going to redraw this angle right over here
- at the center of this protractor, so we can say if this is "Y", than "Z" goes right over here,
- and then the other ray, "ray YX" in this circumstance, will go roughly in that direction.
- And so it is pointing on the protractor to the--let's see this looks like it is the seventieth section,
- this is the eightieth section, so maybe this is, I would guess this is the 77th section,
- so this is pointing to right over here. So when we measure an angle--so we could say that the measure of XYZ
- --assuming I drew it the right way right over here--we could say the measure of
- angle XYZ--sometimes they'll say "angle XYZ is equal to" but this is a little bit more formal-- the
- measure of angle XYZ is equal to 77--and what we do is we call each of these little sections "degrees"--
- so it's equal to 77--sometimes it's written like that, the same way you would write degrees for a temperature outside--
- so you could write 77 degrees like that or you could actually write out the word right over there.
- So each of these sections are in degrees so we're measuring in degrees, but I want to be clear:
- degrees aren't the only way to measure angles. 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", but I'll leave that to another day. 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 this zero degree edge
- of this half circle or protractor, and then I'll put point AB in the--well, assuming I'm drawing it exactly
- the way that it is over there--normally instead of moving the angle you could actually move the protractor
- to the angle--it looks something like that. And you can see that it's pointing to right about--well let's
- just say to right about the 30 degree mark. So you could say the measure of angle BAC is equal to 30 degrees.
- And so you could 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 zero degree angle you have something that's just a closed angle--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-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 doesn't look striaght right-left, but if you rotate it, it would look just like this thing 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, which is a very
- interesting angle that shows up many times in geometry and trigonometry, and there's a special
- word for a 90 degree angle: it is called a right angle.
- So this right over here--assuming if we rotate it around it would look just like this--we would
- call this a right angle. And there is a notation to show that it is a right angle: you draw a little
- kind of part of a box right over there and that tells us that 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 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.
- The two rays--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 has formed an actual line here.
- Z,X, and Y are co-linear. And what we have over here, this is a 180 degree angle, or we should
- say the measure of angle ZXY is 180 degrees. And you can actually go beyond that.
- So if you were to go all the way around the circle, so that you'd get back to 360 degrees, and then you'd
- keep going round and round and round and you'll start to see a lot more of that when you enter
- a trigonometry class. Now there's one last--or two last things that I want to introduce in this video--
- is 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 anything, this is acute,
- 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--let me draw it a little bit better than that--an angle that looks like this,
- and that's one side of the angle, or one of the rays,
- and then I'll put the other one along the base-line right over here. Clearly this is larger than 90 degrees.
- If I were to approximate--let's see that's 100, 110, 120, almost 130--so let's call that maybe a 128 degree angle.
- We call this an obtuse angle. The way I remember it is acute: it's kind of a cute angle.
- It's nice and small. I believe acute, in either latin or greek or maybe both means something like pin or sharp,
- so that's one way to think about it: an acute angle seems much sharper.
- Obtuse, I kind of imagine something that's kind of lumbering and large, so that's how I remember it,
- or you can say it's not acute, it's not nice and small and pointy, so that's one way to think about it.
- But this is just general terminology for different types of angles:
- less than 90 degrees you have an acute angle, at 90 degrees you have a right angle,
- larger than 90 degrees you have an obtuse angle, and if you get all the way to 180 degrees, your angle actually forms a line.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Thank the author
This is great, I finally understand quadratic functions!
Have something that's not a tip or thanks about this content?
This discussion area is not meant for answering homework questions.
At 2:33, Sal said "single bonds" but meant "covalent bonds."
For general discussions about Khan Academy, visit our Reddit discussion page.
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or thanks in Questions
- a question in Tips & Thanks
- an answer that should be its own question