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### Course: High school geometry>Unit 8

Lesson 5: Arc length (from radians)

# Arc length as fraction of circumference

Sal finds the fraction of an arc length out of the entire circumference using the radian measure of the central angle subtended by the arc.

## Want to join the conversation?

• I feel like I missed something... When he says the arc subtends the angle, what does he mean?
(36 votes)
• So radians are proportional to the radius?
(14 votes)
• Think of it like this the "arc that subtends the angle" (aka the part of the circle that you are measuring for the given angle) is a function of BOTH the size of the ANGLE and the RADIUS.

This helped me: Try drawing a little circle inside a bigger circle and then draw some angle that takes a slice out them. Now look at the arc length for both circles: The angle stays the same but the radius is greater on the larger circle and so is its arc length. As the arc length is a function both the angle and the radius. I hope that made sense. It is easy to show on paper.
(40 votes)
• I did it in another way but I'm not sure if it's correct or not. I added 57 degrees to 57 degrees because a radian is equal to 57 degrees right? Then I got 114 degrees, so I divided 114 by 360 and got 0.3166 which a bit close to 1/pi, but I want to know if it's correct or not.
(7 votes)
• If your given information is in radians, then your answer will, unless otherwise specified, be exptcted to be in radians. Intermediate converting from radian to degrees back to radians is introducing unnecessary error to what should be an exact pi radian result.
(12 votes)
• Why are radians "pure"? Is it because they are a universal system in which all being can cooperate to find the perfect measurements in a way we can all understand?
(7 votes)
• If you were an alien on another planet, you would not know what a degree would be, because degrees were invented by humans. Because of this, radians are "pure" because you don't need a measurement system to understand what they mean.
(9 votes)
• I don't understand how Sal derived the arc length from the radian measurement of angle theta. How are the radians of the angle directly related to the radius? Maybe I just missed something, but I'm not understanding how he found the radius or the arc; however, I think it might have to do with C=2(pi)(r)?
(6 votes)
• From the basic definition of radians, the number of radians for a central angle in a circle is the number of "radiuses" that together equal the length of the arc associated with the angle. For example, a central angle that measures 3 radians is associated with an arc whose length is 3 times the radius of the circle. It follows that a full circle is 2pi radians because the circumference (arc length of a full circle) is 2pi times the radius.

Have a blessed, wonderful day!
(7 votes)
• what is a pith?
(2 votes)
• a funny "slang" word that sal came up with to describe
1/pi , like you say 1/3 as "one THIRD" so he said 1 "PI'TH"
hope that helped
(13 votes)
• How come the arc length is 2 radiuses?
(4 votes)
• The definition of an angle that measures 1 radians is that the arc that subtends (lies across from) said angle has to measure 1 radius.

If the arc length across from an angle is less than 1 radius long, then the angle is less than 1 radians.

If the arc length is longer than 1 radius, than the angle is more than 1 radians.

Check out this playlist and lesson plan where Sal gives several examples to learn by and also includes a practice: https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-radians-intro/v/introduction-to-radians

For every 1 radians the arc gets longer by one radius.

Arc Length : Radians

1 radius : 1 radian
2 radii : 2 radians
3 radii : 3 radians
4 radii : 4 radians

... and so on and so forth.
(8 votes)
• I like how Sal says radius + es^x where x = any natural number.
(5 votes)
• So I'm having trouble simplifying this: (5*pi)/6)/2*pi(r). (2*pi(r) is the circumference of the circle. Can someone help me?
(3 votes)
• This is almost like dividing a fraction by a whole number:
(5*pi/6) / (2*pi(r)) = (5*pi)/6) divided by (2*pi(r))
= (5*pi/6) divided by ((2*pi(r))/1)
= (5*pi/6) times (1/(2*pi(r)))
= (5/6) times (1/(2r))
= 5/(12r).

Have a blessed, wonderful day!
(4 votes)
• Why would this arc be equal to 2r? And what exactly is the ratio?
(2 votes)
• The ratio involving arc length is basically saying that the fraction of circumference that is the arc length will be the same as the fraction of the angle made by the arc length over the full 360 degrees or 2pi.

Maybe an example will be better. If the arc length is half of the circumference, then what does that mean the angle will be of the arc length? well, to travel half way around the circle you need half of the full angle around. And it works out for every ratio. a tenth of the circumference will have a tenth of the angle. 1/pi of the circumference will have 1/pi of the angle.

Now, for this, the circumference is 2pir by defenition And the angle is 2 radians. Of course the full angle all the way around is 2pi. So if we call the arc length S that gives us S/(2pir) = 2/2pi.

In english that says the ratio of the arc length S to the full circumference, 2pir is equal to the ratio of the angle of the arc length, 2 radians, over the full angle of the circle, 2pi radians. This would also work for degrees. Anyway, solving it for S gets you 2r like in the video. Let me know if that didn't answer your question though.
(4 votes)

## Video transcript

- Let's say that I have a circle. My best attempt to draw a reasonably perfect circle. So, there you go, not too bad, it's a little bit of a hairy circle but you get the idea. So, this is a circle, this is the center of the circle, and let's say that I have an arc along this circle. So, I'll do the arc in green. So, I have an arc that is part of the circle, and it subtends an angle, so that's my arc. Right over there, and it subtends an angle, and the angle that it subtends, so what I mean subtends, you take each of the endpoints of the arc, go to the center of the circle, go to the center of the circle just like this, and so it subtends angle theta, right over here, so it subtends angle theta, and let's say that we know that angle theta is equal to two radians. So my question to you is what fraction of the entire circumference is this green arc? What fraction of the entire circumference is this green arc? And like always, pause the video, and give it a go. (laughs) All right, so let's think through it a little bit. So, you might say well how do I know that, I don't know what the radius of this thing is, I don't, how do I think through this? And we just have to remind ourselves what radians mean, what radians mean. If an arc subtends the angle of two radians, that means that the arc itself is two "radiuseseses" long. (laughs) So, this right over here, let me make this a little clearer, so this, if the radius is r, if this radius is, I already used that color, if this radius... I have trouble switching colors (laughs) all right. If this radius is length r, then the length, if this angle is two radians, then the arc that subtends it is going to be two radiuses long, so this length right over here, is two radiuses. Now, what fraction of the entire circumference is that? Well, the entire circumference, we know, we know this from basic geometry, the entire circumference is two pi times the radius, or you can say it's two pi radii, two pi "radiuseses", (laughs) two pi radii is the correct way to say it. So, what fraction is it? It's two radii, it's two radii, over two pi radii, over two pi radii, twos cancel out, rs cancel out, and so it is one "pith", (laughs) I guess you could say, it is one over pi of the total circumference.