# Angle measurement and circle arcs | Angles and intersecting lines | Geometry | Khan Academy

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.

Could you maybe post one or two videos a day? I appreciate that you are putting out so much content but I cannot watch it all and it is clogging up my sub box. I do understand that math is very diverse and that you are trying to cover every concept in existence. Although your videos are very helpful, it has become hard to find other videos among the Khan Academy ones.

3 people don't like math I see.

he repeats a word like 10 times lol

helped me a lot thank you

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I swear, this guy doesnt have emotion

So helpful

how to know that is 3/4 or something ???

pls answer me i need ur help

Is this paint?

waste of time!!!!!!!

so helpful thanku

Thank you so much. I learned a lot from this video.

can't you explain this theorm the measure of the angle formed by the lines of two chords intersecting outside a circle is half the difference of the measure of the arc they intercept. it's would be really helpful please explain this

you guessed at the 1/6. exact measurement please. all the other examples were child's play.

thank you

This didn’t help at all

i gess

its

good

is

there

a qwiz

my man, i need the lesson, not the story of the number 360