Question 1

I feel like I missed something... When he says the arc subtends the angle, what does he mean?

Accepted Answer

There is a great visual representation and definition of what it means here:
http://www.mathsisfun.com/definitions/subtend.html

Question 2

So radians are proportional to the radius?

Accepted Answer

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.

Question 3

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.

Accepted Answer

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.

Question 4

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?

Accepted Answer

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.

Question 5

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)?

Accepted Answer

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!

Have a blessed, wonderful day!

Question 6

How come the arc length is 2 radiuses?

Accepted Answer

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.

Question 7

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?

Accepted Answer

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!

Question 8

what is a pith?

Accepted Answer

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

Question 9

Why would this arc be equal to 2r? And what exactly is the ratio?

Accepted Answer

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.

Question 10

Confused dont know weather to muiltipy or divide by 2 pie any more?

Accepted Answer

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.

High school geometry

Course: High school geometry > Unit 8

Arc length as fraction of circumference

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