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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.

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  • old spice man green style avatar for user Jonathan Ziesmer
    I feel like I missed something... When he says the arc subtends the angle, what does he mean?
    (36 votes)
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  • old spice man green style avatar for user Jonathan Ziesmer
    So radians are proportional to the radius?
    (14 votes)
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    • blobby green style avatar for user Nathan Sturgess
      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)
  • purple pi purple style avatar for user Alex Hickens
    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)
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  • leaf green style avatar for user Hana Vitt
    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)
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  • piceratops ultimate style avatar for user dollyrauh
    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)
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    • primosaur seed style avatar for user Ian Pulizzotto
      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)
  • hopper cool style avatar for user AksharLEO
    what is a pith?
    (2 votes)
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  • male robot hal style avatar for user Universe14
    How come the arc length is 2 radiuses?
    (4 votes)
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  • hopper cool style avatar for user \«π
    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)
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  • duskpin ultimate style avatar for user sanaa
    Why would this arc be equal to 2r? And what exactly is the ratio?
    (2 votes)
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    • female robot grace style avatar for user loumast17
      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)
  • old spice man green style avatar for user lola gerguis
    Confused dont know weather to muiltipy or divide by 2 pie any more?
    (2 votes)
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    • leaf green style avatar for user The Riemann Hypothesis
      you divide by 2 pi. This is because the expression takes the form (theta)/2 pi radians. You would say 120 degrees is 1/3 of 360 degrees because 120/360 is 1/3. Fractions take the form (part of a whole) / (whole). Here, 2 radians is the part of the whole, and 2 pi radians is the whole. So, you end up with (2 radians) / (2 pi radians), and the 2 and the radians cancel out, leaving you with 1/pi of a circle. I hope this was helpful.
      (3 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.