Topic B: Lessons 9-10: Arc length and sector area
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-- should be the same as the ratio 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. 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. So you have 10 degrees over 360 degrees. So we could simplify this by multiplying both sides by 18 pi. And we get that our arc length is equal to-- well, 10/360 is the same thing as 1/36. So it's equal to 1/36 times 18 pi, so it's 18 pi over 36, which is the same thing as pi/2. So this arc right over here is going to be pi/2, whatever units we're talking about, long. Now let's think about another scenario. Let's imagine the same circle. So it's the same circle here. Our circumference is still 18 pi. There are people having a conference behind me or something. That's why you might hear those mumbling voices. But this circumference is also 18 pi. But now I'm going to make the central angle an obtuse angle. 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 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, a, and the circumference of the entire circle, 18 pi, should be the same as the ratio between 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. We get a is equal to-- this is 35 times 18 over 36 pi. 350 divided by 360 is 35/36. So this is 35 times 18 times pi 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 35/2 pi. Let me just write it that way-- 35 pi over 2. Or, if you wanted to write it as a decimal, this would be 17.5 pi. Now does this makes sense? This right over here, this other arc length, when our central angle was 10 degrees, this had an arc length of 0.5 pi. So when you add these two together, this arc length and this arc length, 0.5 plus 17.5, you get to 18 pi, which was the circumference, which makes complete sense because if you add these angles, 10 degrees and 350 degrees, you get 360 degrees in a circle.