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Circles, Arcs, and Sectors
Arcs and Sectors. In the previous lesson, we said that arc measure, was one way to talk about the size of an arc. So the number of degrees, that an arc has is one we can say how big an arc is. As long as we’re looking at two arcs, in the same circle. Arc measure is perfectly adequate, to compare the size of two different arcs.
If we start comparing arcs in different size circles, it becomes clear that arc measure alone is not sufficient. The two arcs here, the arc BC, and the arc CD. Those are both 30 degree arcs, because they share a central angle of 30 degrees. So, they have the same measure, but clearly, they’re not identical. They’re not the same in every way.
They have different lengths. If these were actual paths outside, it would take us longer to walk from C to D, than it would take us to walk from A to B. Those paths have two different lengths. So the arc measure is more about the curvature of the arc, but we also have to consider the length.
So then it becomes a question. How do we find the length of an arc? Well first of all, this length is called an arclength, and it is found by setting up a proportion, a part-to-whole proportion. So, I’m very much gonna emphasize here, don’t simply memorize a formula, I want you to understand the logic of it.
We’re really asking the question, how much of this circle do we have? And, on the basis of that question, we’re setting up this proportion. So, the arclength is part of the circumference. And, so you compare the arc to the circumference. That’s the part to the whole on one side of the equation. Similarly the arc measure or the measure of the central angle, those two are interchangeable, they’re the same, is the part of 360 degrees all the way around the circle.
So the proportion we set up, is arclength over 2pi r equals angle over 360. For example, in this circle, we have an angle of 120 degrees. Well, 120 degrees is one third of 360, so we have a third of the whole circle here. And because that angle is a third of 360. The arc has to be a third of the circumference.
The circumference we can figure out easily enough, that’s pi r squared, sorry, 2pi r, which is 48pi, because the radius is 24, and then one third of that is 16pi. So we have an arclength of 16. Here’s a very easy practice problem. Pause the video, and then we’ll talk about this.
Okay. If the length of the arc is 12pi, then find the area of the circle. Well we’re given the angles, so the first thing we’re gonna do is figure out how much of the circle do we have? We gonna put that angle over 360.
And 72 divided by 360 if you think about it, 72 is 2 times 36, so we have 2 times 36 over 10 times 36, 2 over 10 which is one fifth. That’s a handy fraction to know, to know that 72 degrees is one fifth of 360. So we have a fifth of a circle. And so that means that, that arc is one fifth of the circumference. So the circumference must be 5 times that, or 60.
60pi. And that 60pi has to equal 2pi r. Here’s one of our big circle strategies. Use the information you’re given to find the radius. Once you find the radius, you can find everything else you need. So we divide, we get the radius equals 30.
Then of course the area equals pi r squared equals 900pi. That is the area of the circle. Just as we can divide the circumference into fractions, so we can divide the whole area of the circle. A slice of the circle like this is called a circular sector. Some people like to think of it, as a slice of pie, or a slice of pizza.
To find the area of a circle, we set up another part-to-whole proportion, in which the area of the sector is part. Of the area the whole circle pi r squared. So again, don’t simply memorize a formula. Think about the logic of this. How much of the circle do we have?
So we setup the, the ratio for the angles, the ratio of the areas and we set them equal. So area of circle over pi r squared, would equal the angle over 360. For example, we’re given a 60 degree angle. Well 60 degrees, 60 over 360 that’s obviously one sixth.
We’re dealing with one sixth of the circle. Well, if we want that area, the whole area, the area of the whole circle, of course, is pi r squared, we could write that as pi times 18 times 18. Now I’m not gonna make the mistake of multiplying that out, I’m just gonna leave that as 18 times 18. Because I’m gonna take one sixth of that.
And when I take one sixth of it, I divide one of those 18s, I divide it down, 18 divided by 6 is 3, so then I just have to really multiply 3 times 18, that’s 54pi, and that is the area of the sector. Here’s a practice problem. Pause the video and then we’ll talk about this. So we’re given the warning, that the diagram is not drawn to scale.
We don’t actually know, what that means at the beginning, okay. The circle has a radius of 8, all right? So, we could figure out the area of the circle. And the sector has an area of 16pi. Well, that’s interesting. From the radius, we could figure out the area of the whole circle, 64pi.
Now let’s figure out how much of the circle we have, 16pi over 64pi, that’s one-forth. So, that means that the angle JOK, that’s a 90 degree angle. Okay. So that’s where the diagram was fooling us. If that angle were drawn to school, that act, would actually be a right angle, JOK.
Okay, but now we know that’s a right angle. And so now, we want the, the length of arc now, this is tricky. JLK, so that’s not the little arc that runs along the sector, that’s the big wrap around arc, that starts at J, goes counterclockwise around through L, and arrives at K. So that would be three quarters of the circle, three quarters of the circumference.
Well the circumference we can figure out easily enough is 16pi. Three quarters of that circumference would be 12pi. And that is the length of that gigantic arc. In summary, we find arclength and area of a sector, by setting up part to whole proportions. So we’d say arclength over the circumference 2pi r equals the angle over 360.
Again we’re just figuring out how much of the circle we have, on each side of the equation. Or we set up a similar ratio, area of a sector over pi r squared, also equals angle over 360. We’re figuring out how much of the circle do we have.
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