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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. That's one way we can say how big an arc is. As long as we're looking at two arcs in the same circle, arc measures 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 pads 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 find the length of an arc? Well, first of all, this length is called an arc length, and it is found by setting up a proportion, a part-to-whole proportion. So, I'm very much going to 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 the 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 proposed we set up is arclength/2r = angle/360. For example, in this circle, we have an angle of 120 degrees, well 120 degree is 1/3 of 360. So we have 1/3 of the whole circle here, and because that angle is 1/3 of 360, the arc has to be 1/3 of the circumference.

Well, the circumference will get figured out easily enough, that's 2r, which is 48, because the radius is 24, and then 1/3 of that is 16, so we have an arc length 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 12, then find the area of the circle.

Well, we're given the angle, so the first thing that we're gonna do is figure how much of the circle do we have, and we're gonna put that angle over 360, and 72/360, if you think about it 72 is two times 36, so we have two times 36 over 10 times 36. 2/10, which is 1/5. That's a handy fraction to know, to know that 72 degrees is 1/5 of 360.

So I have 1/5 of the circle, and so that that means that that arc is 1/5 of the circumference. So the circumference must be 5 times that, or 60, and that 60 has to equal 2r. Well 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 r2 = 900. 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 of the whole circle, r2. So again, don't simply memorize a formula, think about the logic of this. How much of the circle do we have?

We set up the ratio for the angles, the ratio of the areas, and we set them equal. So, area of a circle/r2 = angle/360. For example, if we're given a 60 degree angle, well, 60 degrees, 60/360 that's obviously 1/6, we're dealing with 1/6 of the circle. Well, if we want that area, the whole area, the area of the whole circle, of course is r2, we could write that as (18)(18).

Now, I'm not gonna make the mistake of multiplying that out, I'm just gonna leave that as (18)(18), because I'm gonna take 1/6 of that, and when I take 1/6 of it I divide one of those 18's, I divide it down 18/3 = 3, so then I just have to really multiply (3)(18)=54, 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 a warning, 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 16. Well, that's interesting.

From the radius, we could figure out the area of the whole circle, 64. Now let's figure out how much of the circle we have, 16/64 = 1/4. 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 scale, that would actually be a right angle JOK. Okay, but now we know that's a right angle, and so now, we want 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 16. Three quarters of that circumference would be 12, and that is the length of that gigantic arc. In summary, we find arclength and an area of a sector by setting up part-to-whole proportions. So we'd say arclength over the circumference, 2r = angle/360.

Again, we're just figuring out how much of the circle do we have on each side of the equation, or we set up a similar ratio, area of the sector/r2 = angle/360. We're figuring out how much of the circle do we have.

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