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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.

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• I feel like I missed something... When he says the arc subtends the angle, what does he mean?
• 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.
• 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.
• 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?
• 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.
• 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)?
• 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!
• what is a pith?
• 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
• How come the arc length is 2 radiuses?
• 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

... and so on and so forth.
• 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?
• 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!
• Why would this arc be equal to 2r? And what exactly is the ratio?
• 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.