Transcription
Hello everybody, how you doing? This is Mr. D. On this video, I'm going to talk about a sector of a circle. Well, what is a sector? A sector is part of the area of a circle bound by the central angle and its intercepted arc. What do I mean by central angle? What do I mean by intercepted arc? Let me explain these two before I go any further.
A central angle is an angle, uh, that uses the center of the circle as the vertex of the angle. So we've got this angle here from B to A to C, and notice how the center of the circle, A, is also the vertex of the angle. Uh, and then so, well, the intercepted arc is the arc that this central angle creates. Basically, if I look, this central angle B A is hitting the circle at B and at C. So the intercepted arc is the arc created between those two points around the circle. So the sector is part of the area of a circle bound by the central angle and the intercepted arc. Basically, it's everything in between the central angle and the arc that the central angle created. And so it's the area of this portion of the circle. So it's part of the area. Uh, and so I think of this whenever it comes to food, like, uh, like pizza. If I have a slice of pizza and I have a slice of pizza from B to A to C, and I were to pull the slice out, well, how much area is that slice of pizza? Uh, whenever I get pizza, I generally grab the largest slice because that's just what I want to do. And so I'm grabbing the slice with the largest sector area. Uh, and so that might make a little bit more, more sense to you.
But how do you find the area of a sector? Well, you need a central angle and you need a radius. And if you know the central angle and a radius, you can use a proportion to cross-multiply and find the area of that sector. Uh, and so what we have here, let's look at this. This proportion. This proportion is comparing the central angle of the sector divided by 360 degrees, equal to the sector divided by the area of this circle. And so again, pi r squared is the area of a circle. So what we're doing here, essentially, is comparing part of the area that we're trying to find over the entire area of the circle, setting that equal to the central angle of that sector over 360 degrees. We're comparing part over whole and setting it equal to part over whole. These are quantities that we can cross-multiply. And if we know central angle and radius, we can use to find the sector or the area of the sector.
So let's do two example problems. First one's pretty easy. Second one's a little bit more challenging. So I recommend paying attention to the second one. Uh, find the area of the shaded region. Well, shoot, what is the area of the shaded region? Well, shoot, this is the sector here. This is what we're trying, this is what we're talking about here. Um, I'm trying to find this, so I'm going to call this X. This is going to be the unknown quantity here. Now, do I know a central angle? I do. Do I know the radius? I do. So I should be able to plug in the numbers here and then solve the problem and find the sector. So let's set up the proportion. Do I know the central angle? I do. It's 75 degrees. Divide that by 360 degrees equals, uh, the sector. Well, I'm calling this X for this problem. Uh, I know what pi is. Pi is 3.14, and the radius is 10 inches. I'm going to keep the units in here. That's going to matter. So we're going to square this. So this is pi r squared. Uh, now I'm going to simplify the pi r squared here, uh, before I go any further. And if I type in my calculator, 3.14 times 10 squared, I get 314. And so what we're going to have here on the bottom portion is 314 inches squared. When I square the 10, I'm also squaring the inches. And so we're going to have inches squared on the bottom here. Sector is still X. 75 degrees on top, 360 degrees on the bottom. Well, I'm going to ignore these degrees right now. Those don't really matter to me, but the units do. And so, uh, we're going to, I'll explain that here when I get done with the problem. I'm going to cross-multiply. What is 75 times 314? 75 times 314, I get a huge number, 23,550. And this is in inches squared. We've got to keep these units going with us here. So this is inches squared. This is equal to 360x. Now my goal is to get that three, sorry, that X by itself. So I'm going to divide both sides by 360. And when I do this, what is this, uh, divided by 360? I'm going to round this to the nearest tenth. I have 65.4 inches squared equals X. Now, why am I holding on to this inches squared? We're finding area. Area is always going to have some units squared. And so this guy here is the area of this portion of the circle. So this is the sector. The area of this shaded region is the area of the sector. And so again, inches squared is going to be important here because we're finding area. Area is going to always have some units squared to go along with it. And I can just see somebody on a multiple choice test having 65.4 inches and 65.4 inches squared, and you need to know since this is area, you need to have squared.
I'm going to do one more problem and then call it quits. And so, um, find the radius of the circle if the area of the shaded region is 50 pi. Um, so 50 pi, what is the shaded region? Okay, this is actually the sector. This is the area of the sector here, so that's going to be important here. So we know the sector. We know the central angle is 110. We're trying to find the radius. How do we do this? Uh, and so let's, let's, let's plug in the, uh, numbers into the equation, see what we come up with. I have 110 degrees, uh, divided by 360 degrees. So that's kind of nice. Equals the sector is now 50 pi divided by pi r squared. Now, R is what I'm trying to find. My goal is to find the radius. And so I'm going to leave this as an R here. I'm not going to make this X or anything. Now, I'm going to ignore the degrees here. That's not going to do anything for me. Um, this problem actually doesn't have any units, so that's kind of nice. No inches or feet or anything. Um, now if I look here very carefully, I can actually simplify the right half of this equation. If I have pi on top and pi on the bottom, these guys can simplify and cancel out. And if you don't understand what I'm talking about, I basically have 50 times 3.14 and then I have 3.14 times radius squared. Well, anything over itself will cancel out. And so that's going to simplify this problem down quite a bit. And so that 50 pi isn't really something to be afraid of. So I'm going to keep this 110 divided by 360 equals 50 over r squared. Oops, that does not look like an r squared. Sorry, I'm just clarifying my handwriting here. So we're going to cross-multiply. Likely normally would. 110 times r squared is 110 r squared. Now, what's 360 times 50? Uh, I get 18,000. Pretty big number, right? So now we need to get the r squared by itself to get radius by itself eventually. Divide both sides by 110. And I have the radius squared is equal to, plugging in my calculator, I get 163.6363 keeps going on forever. I'm going to round this to the nearest tenth, 163.6. It shouldn't have, uh, matter here. I hope the rounding here. So again, I rounded this to the nearest tenth. Now, am I done here? No, no, no, no, no, not quite. This is radius squared. I need the radius. Well, how do you cancel out radius squared? You've got to square root. So I'm square rooting both sides here. Square root both sides here. This will cancel out the squared next to the R. So the radius is equal to whatever 163.5, the square root of this is. So let's take our calculator. Square root of 163.5. And I get. Have a good day. Bye-bye.