 As we finish lecture 7, let's reflect back on where we came from, right? We've observed that the arc length on a circle is directly proportional to the angle measure that is the measure of the central angle associated to it. And so that's where we get this formula s equals r theta, where the arc length and the angle are proportion to each other with respect to the radius of the circle. And it's imperative that when you use this formula that the angle measure be in radians. It has to be in radians because radians are utilizing this fact that angle measure and arc length are proportional to each other. Now it turns out we can extend this result to one other setting, right? We have this picture right here, which if you have the center of the circle, two points on the circumference here, and then the arc length, the distance between them along the circumference there. But what if we look at this angle here, the central angle, and we ask ourselves how much area fills in this sector, this so-called pizza slice? We want to find a formula for the area of the sector of a circle. And again, we're going to use a proportionality argument to find this formula, to put this together, right? Because the idea is have you used a much larger angle like something like this, the area is going to be larger, and it's going to be proportional using the uniformity of the circle, the larger the angle, the larger the area. So angle measure and area are going to be proportion to each other as well. And so setting up that proportion, we're going to look at something like the following. Let's take area of a specific sector, like we'll take this pizza slice right here, and divide it by its angle theta. That will be equal to pi times r squared over two pi. So what's going on here is that if you take the area of the entire circle, that is going to be pi r squared. And if you take the angle measure of the entire circle, that's a complete rotation, that's going to be two pi radians. And so by proportionality, we get the following formula, A over theta equals pi r squared over two pi. We're here, we're using radian measure. Now simplifying this thing, of course, we can divide out the pi. So you get A over theta is equal to r squared over two. And if you times both sides by theta, we end up with the following formula right here. The area of the sector of a circle will equal one half r squared where r is the radius times theta, where theta is the radian measure of the central angle that produces this sector. And so let's show you how we can use this formula here in practice. Find the area of the sector formed by a central angle of 1.4 radians with a radius of 2.1 meters. So just using the formula we had above, area equals one half r squared theta. And so plug in the things we know. We know the radius of the circle is going to be 2.1 meters. So we get 2.1 meters squared. Notice, as we've talked about before, that radian measure is kind of considered a unitless measurement. When you do formulas, radians don't change the units whatsoever. But if you take a distance like r and you square it, they're going to get square units. So if the radius of the circle is measured in meters, then the area of the sector should be meters squared. You're going to see that, the meter squared right here. The angle measure, like we were told, is 1.4 radians. So we're going to get 1.4 right here. And then we just, of course, just have to simplify this thing now. 1.4 divided by 2. So if you put these two friends together, you're going to end up with a 0.07. If you take 2.1 and you square it, you're going to end up with 4.41. Our units, of course, are going to be meters squared here. 4.41 times 0.07. You know, feel free to use a calculator to help you with this arithmetic here. You're going to get 3.087 meters squared. So using the formula, we can plug and chug and find the measurement of the area of this sector, given the radius and the angle, if the angle is in radians. Let's consider another problem. If the sector formed by a 15 degree angle has an area of pi over 3 square centimeters, what was the radius of the circle? So notice what's going on here. With our formula, area equals 1.5 radias squared theta. So we know theta is this 15 degrees. We know the area we want to solve for the radius for which we can do that. I mean, we can just see symbolically how one would do that. Times both sides of the equation by 2 to cancel out the 1.5 here. Divide both sides by theta. You're going to end up with, of course, that r squared is equal to 2a over theta. Take the square root because the radius does have to be a positive here. You're going to get the r equals the square root of 2a over theta. So we can solve for r easy enough if we know a and theta. The area, of course, was given to us. We end up with pi over 3. The angle measure does have to be in radians. So that's critical here. So if we take 15 degrees, which is our angle measure theta here, to switch it to radians, we need to take pi over 180 degrees so that the degree measure cancels out. And then we end up with just radians here now. And so with radians, you end up with this 15 pi over 180. Certainly, 5 goes into 15 and into 180. You also get that 3 goes into both of them. So this would actually simplify to give you pi 12. And so we can then put that into our formula right here. But we have a bunch of fractions inside of fractions. So before we go any further, we need to clean this thing up. And so doing so, we end up with a, well, let's write it in pi over 12, like so. So we have fractions divided by fractions. We have two pi thirds multiplied by the reciprocal. You're going to get 12 over pi. And this is all inside the square root. Let's try to clean it as much as it's up before we start throwing this to a calculator. The pi's are going to cancel out like so. 3 goes into 12 four times. And so you get 2 times 4. So we could write this as the square root of 8. That would be acceptable. But as 4 is a perfect square, you take the square root of 4, you're going to get a 2. This is the same thing as 2 root 2. If you want an exact answer, square root of 8, same thing. Or as it's probably inappropriate to get an approximate answer for this, let's take the square root of 8. And we're going to get 2.828. What are the units in play here? Well, since the area was measured in centimeters, that means we should, well, centimeters squared, I should say. That means the radius of the circle will be measured in centimeters as well. So we get that the radius of our circle here is approximately 2.828 centimeters. Let's look at one more example here. And this one we could argue is like a story problem, right? A lawn sprinkler located at the corner of a yard is set to rotate through 90 degrees and project water out 30 feet. So think about this little little sprinkler for a moment. It's here in the corner. It shoots out a 90 degree angle. So maybe it's like in the corner of our lawn here. And it does have a circular arc right here, which it shoots it out 30 feet like so. And this measurement is a 90 degree angle, which already knowing what's going to happen here, we just switch this to radians. You're going to get pi halves as the radiant measure of our angle theta right here. What area of the lawn is watered by this sprinkler? So if you think about how many cubic feet of lawn is watered right here, well, by the area formula A equals one half r squared theta for which we get the radius here is 30 feet. How far does the sprinkler shoot? 30 feet there. We're going to square that. And then the angle measure, it is a corner. So it gets a whole 90 degrees, but we have to convert it to radians. So we get pi over two, like so. And then let's try to simplify this thing. One half times pi halves, of course, it's going to be pi fourths. If you take 30 squared, you're going to end up with 900 and 100 feet squared. Of course, four goes into 900, 225 times. So 225 pi times feet squared. To make this practical, we probably don't want the pi. We probably need to have it as some decimal approximation. We'll round to the nearest foot. If you take 225 and times it by pi, this will be approximately 707 square feet, which is how much lawn is watered by this sprinkler right here. So thanks for watching. And this is the end of lecture seven, which we learned about radiant measure and some applications of radiant measure. If you learned something, please hit the like button. If you want to see more videos like this in the future, subscribe to the channel. And as always, if you have any questions, feel free to post them in the comments below and I'll be glad to answer them.