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CCSS.Math:

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 arc length arc length to the entire circumference circumference should be the same as the ratio of the central angle 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 so 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 you have 10 degrees over 360 degrees over 360 360 so you could we can simplify this by multiplying both sides by 18 PI multiplied both sides by 18 PI and we get that our arc length our arc length is equal to well 10 over 360 is the same thing as 1 over 36 so it's equal to 1 over 36 times 18 pi which is just so it's 18 PI over 36 which is the same thing as PI over 2 so this arc right over here is going to be PI over 2 whatever units were talking about long now let's think about another scenario let's imagine another or the same circle the same circle here our circumference is still 18 pi circumference is still 18 5 are people having a conference behind me or something that's why you might hear that those mumbling voices but this circumference is also 18 pi so this is also 18 pi but now I'm going to make the central angle I'm going to make it an obtuse angle so the central angle now 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 eye 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 the ratio between our arc length a and the circumference of the entire circle 18 pi should be the same should be the same as the ratio between the and our central angle 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 multiplied both sides by 18 pi we get a is equal to let's see this is 35 times 18 over 36 PI so this is so 30 350 divided by 360 is 35 over 36 so this is 35 times 18 times pi over 36 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 we are left with 35 over 2 pi or we could even say let me just write it that way 35 35 PI over 2 thirty-five PI over two or if you wanted to write as a decimal this would be 17.5 pi now does this make sense this right over here the this other arc length when our central angle is 10 degrees this had an arc length of 0.5 pi so when you add these two together this arc length in this arc length 0.5 plus 17.5 you get to 18 pi which was a circumference which makes complete sense because if you add these angles 10 degrees and 350 degrees you get 360 degrees in a circle