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## High school geometry

# Arc length from subtended angle

CCSS.Math:

Finding the length of an arc using the degree of the angle subtended by the arc and the perimeter of the circle. Created by Sal Khan.

## Want to join the conversation?

- I am confused about 18 pi as well. Aren't all circles 2 pi? Does 18 pi mean he is going around the circle 9 times?(63 votes)
- 2 pi radians is 360 degrees, so yes, all circles have an
**angle**of 2 pi. In this video however, Sal is talking about the**length**of the circumference or a fraction of the circumference. This value will depend on the size of the circle.(140 votes)

- What does subtends mean(60 votes)
- If an arc subtends a particular angle at the centre of the circle, it means that if you draw straight lines from each end of the arc to the centre of the circle, that's the angle you get between the two straight lines.(65 votes)

- I don't understand where 18 pi came from?(4 votes)
- Hi Sara. 18π is the circumference of the circle and was given to us in the problem statement.(45 votes)

- How would you solve this problem if you were given the radius of the circle instead of the circumference?(11 votes)
- In that case, all you have to do is multiply the radius by two, to get the diameter. Then, multiply the diameter by pi to get the circumference.(13 votes)

- how do you turn a fraction into degrees without it being a decimal?(9 votes)
- You can do proportions. You multiply the denominators, then divide the first numerator by that number. Voila! You get the answer, no decimals included.(10 votes)

- At2:35there is a pop up that says" Sal said"obtuse angle" but meant "reflex angle." What is a reflex angle?(2 votes)
- A reflex angle has a measure between 180 and 360 degrees.(16 votes)

- At about4:00, couldn't you just subtract pi/2 from 18pi, because 360 degrees minus 350 degrees is 10 degrees? Wouldn't that save a lot of time?(4 votes)
- You could, but I think Sal just did it the long way to demonstrate how its done.(2 votes)

- What if there is a question that says to find the arc length, but provides no information on the circumference, instead it provides information on the radius. I have a question like this on khan academy.(3 votes)
- Given a circle's radius, you can find its circumference as 2π times the radius.(3 votes)

- wait a minute isn't the arc the same as 350?(3 votes)
- Well no because it is the length not the measure(3 votes)

- how about if the central angle is a fraction?(2 votes)

## Video transcript

I have a circle here whose
circumference is 18 pi. So if we were to measure all
the way around the circle, we would get 18 pi. And we also have a
central angle here. So this is the
center of the circle. And this central angle
that I'm about to draw has a measure of 10 degrees. So this angle right
over here is 10 degrees. And what I'm curious
about is the length of the arc that subtends
that central angle. So what is the length of
what I just did in magenta? And one way to think about
it, or actually maybe the way to think about it, is
that the ratio of this arc length to the entire
circumference-- let me write this
down-- should be the same as the ratio
of the central angle to the total number
of angles if you were to go all the way around the
circle-- so to 360 degrees. So let's just think about that. We know the
circumference is 18 pi. We're looking for
the arc length. I'm just going to call
that a. a for arc length. That's what we're going
to try to solve for. We know that the central
angle is 10 degrees. So you have 10 degrees
over 360 degrees. So we could simplify this
by multiplying both sides by 18 pi. And we get that our arc
length is equal to-- well, 10/360 is the same
thing as 1/36. So it's equal to
1/36 times 18 pi, so it's 18 pi over 36, which
is the same thing as pi/2. So this arc right
over here is going to be pi/2, whatever units
we're talking about, long. Now let's think about
another scenario. Let's imagine the same circle. So it's the same circle here. Our circumference
is still 18 pi. There are people
having a conference behind me or something. That's why you might hear
those mumbling voices. But this circumference
is also 18 pi. But now I'm going to make the
central angle an obtuse angle. So let's say we were to
start right over here. This is one side of the angle. I'm going to go and
make a 350 degree angle. So I'm going to go all
the way around like that. So this right over here
is a 350 degree angle. And now I'm curious
about this arc that subtends this really huge angle. So now I want to figure out this
arc length-- so all of this. I want to figure out this
arc length, the arc that subtends this really obtuse
angle right over here. Well, same exact logic-- the
ratio between our arc length, a, and the circumference of
the entire circle, 18 pi, should be the same as the
ratio between our central angle that the arc subtends, so
350, over the total number of degrees in a
circle, over 360. So multiply both sides by 18 pi. We get a is equal to-- this
is 35 times 18 over 36 pi. 350 divided by 360 is 35/36. So this is 35 times
18 times pi over 36. Well both 36 and 18
are divisible by 18, so let's divide them both by 18. And so we are left with 35/2 pi. Let me just write it
that way-- 35 pi over 2. Or, if you wanted to
write it as a decimal, this would be 17.5 pi. Now does this makes sense? This right over here,
this other arc length, when our central
angle was 10 degrees, this had an arc
length of 0.5 pi. So when you add these two
together, this arc length and this arc length,
0.5 plus 17.5, you get to 18 pi, which was
the circumference, which makes complete sense because
if you add these angles, 10 degrees and 350 degrees, you
get 360 degrees in a circle.