WEBVTT 00:00:04.354 --> 00:00:10.604 In this video, we look at some applications of radian measure. We'll start with the 00:00:10.609 --> 00:00:18.368 area of a sector of a circle. So a sector of a circle is like a pie piece where the point 00:00:18.370 --> 00:00:22.620 is at the center of the circle and the two sides here are both radii. 00:00:22.622 --> 00:00:28.371 We're going to start by looking at the fraction of the circle represented by that sector. 00:00:28.373 --> 00:00:39.135 We could calculate this two different ways. If we knew the area of the sector, 00:00:39.137 --> 00:00:48.488 we can write that over the area of the entire circle, and that's going to tell us 00:00:48.490 --> 00:00:53.887 the fraction of the circle that the sector represents. We could do this another way. 00:00:53.889 --> 00:01:14.000 We could look at angles. So we're going to call this central angle of the sector theta, 00:01:14.500 --> 00:01:18.430 and if we look at the angle that would be if we went all the way around the circle, 00:01:18.432 --> 00:01:26.179 then that's one complete revolution or 2 pi radians. Either way we calculate the 00:01:26.181 --> 00:01:30.124 fraction of the circle present. It's the same fraction of the circle so these 00:01:30.126 --> 00:01:36.856 two ratios are equal. Well, we know a little bit more than what I've written. 00:01:36.858 --> 00:01:41.856 The area of this sector is what we'd like to find. Let's just call that A. 00:01:41.858 --> 00:01:47.710 The area of the entire circle, we can calculate that using an old formula from geometry. 00:01:47.712 --> 00:01:59.964 It's pi times the radius squared. Our radius is r. The central angle, theta, and then 2 pi. 00:01:59.966 --> 00:02:04.228 One thing I should point out is it's pretty important theta is in radians, 00:02:04.238 --> 00:02:10.226 since I've used 2 pi radians to represent the entire circle, all the way around it. 00:02:10.228 --> 00:02:21.725 Ok, so let's solve for A, the area of the sector. Multiply both sides by pi r squared. 00:02:21.727 --> 00:02:27.746 We can divide out those pi's and then sort of rearrange what's left. 00:02:27.746 --> 00:02:39.736 The area of the sector is 1/2 the radius squared times theta, with theta in radians. 00:02:39.738 --> 00:02:42.737 Now let's practice using this formula. 00:02:44.177 --> 00:02:49.272 In this example, we're asked to find the area of a sector of a circle where the 00:02:49.274 --> 00:02:53.524 central angle is 40 degrees and the radius is 5 centimeters. So we've just found 00:02:53.526 --> 00:03:00.270 this great formula. The area is 1/2 r squared theta, but theta needs to be in radians. 00:03:00.272 --> 00:03:05.788 So that's our first problem. So, whenever we're using a formula, we should always 00:03:05.790 --> 00:03:11.042 write it down before we begin, and then we can substitute in there. But as we 00:03:11.044 --> 00:03:18.052 discussed, this needs to be in radians. So we're going to take our 40 degrees 00:03:18.054 --> 00:03:27.555 and to the side here convert that to radians. How do we convert? 00:03:27.557 --> 00:03:36.811 Pi radians is 180 degrees, so we're going to multiply this by pi over 180 degrees. And 00:03:36.813 --> 00:03:43.074 for now, we're just going to reduce that to lowest terms and keep it as an exact value, 00:03:43.076 --> 00:03:47.082 so I can go a little bit further. That's really 2 pi over 9. 00:03:47.084 --> 00:03:57.337 Now we're ready to substitute. The area is 1/2 the radius, which is 5, squared 00:03:57.339 --> 00:04:06.100 times 2 pi over 9. I should stop here because my instructions didn't really tell me 00:04:06.102 --> 00:04:10.102 how to leave my answer. So we're going to start by giving an exact answer 00:04:10.102 --> 00:04:14.608 that's left in terms of pi, and then we'll approximate it if we need to later. 00:04:14.610 --> 00:04:28.100 Alright, so what can we do? Well 1/2 times 2 is 1, so I can write this as 25 pi over 9. 00:04:28.102 --> 00:04:32.620 What kind of units would the area of the sector have? 00:04:32.622 --> 00:04:37.623 The radius is given in centimeters, so area should be in square centimeters. 00:04:37.625 --> 00:04:48.636 This is an exact version of our answer. Sometimes you're asked to approximate 00:04:48.638 --> 00:04:52.140 your answer to one or two decimal places. At that point you would take your 00:04:52.142 --> 00:04:57.648 calculator out and multiply and divide on your calculator to get an approximation. 00:04:57.650 --> 00:05:04.406 Let's do that real quick. 00:05:04.408 --> 00:05:12.397 This is approximately 8.73. So we're going to use 2 decimal places here, 00:05:12.399 --> 00:05:17.143 centimeters squared. The number of decimal places will always be told to you 00:05:17.145 --> 00:05:20.500 in your homework or in any problem that you're working. 00:05:22.926 --> 00:05:29.675 Here's a second example of the area of a sector of a circle. Find the radius 00:05:29.677 --> 00:05:34.190 of a sector of a circle given the central angle is 3/5 radians and the area of the 00:05:34.192 --> 00:05:40.684 sector is 600 square feet. Round our answer to one decimal place. 00:05:40.686 --> 00:05:47.937 So we know we can use that formula. Area is 1/2 r squared theta for a sector of a circle. 00:05:47.939 --> 00:05:52.397 Theta needs to be in radians, so let's check that first. Great, this time we have 00:05:52.399 --> 00:05:57.460 our angle given in radians. But what's the difference in this problem? 00:05:57.462 --> 00:06:01.899 I'm not looking for the area. I already know that's 600 square feet. 00:06:01.901 --> 00:06:05.969 I know the central angle, but I need to find the radius. So the algebra 00:06:05.971 --> 00:06:09.717 in solving this one is a little bit more involved. So let's start by 00:06:09.719 --> 00:06:15.965 substituting what we know. The area of the sector is 600. The radius 00:06:15.967 --> 00:06:22.232 is what I'd like to try to find, and theta is 3/5 radians. 00:06:22.234 --> 00:06:26.480 I'm going to simplify the right hand side by multiplying those fractions. 00:06:26.482 --> 00:06:36.231 That's 3/10 times r squared, and I need to get that r squared by itself. 00:06:36.231 --> 00:06:42.478 So let's multiply both sides by the reciprocal of 3/10. That would be 10/3. 00:06:42.480 --> 00:06:56.762 So we have 10/3 times 600 equals, 10/3 times 3/10 is 1, so this is just r squared. 00:06:56.764 --> 00:07:05.476 10 times 600 is 6000, divide that by 3, so we have 2000 equals r squared. 00:07:05.478 --> 00:07:09.479 Now that's not a perfect square, so I'm going to grab my calculator 00:07:09.481 --> 00:07:18.790 so we can approximate it. r is going to be the square root of 2000. 00:07:18.792 --> 00:07:31.291 It's certainly going to be a positive number because it's the radius of our circle. 00:07:31.293 --> 00:07:40.313 The square root of 2000 is about 44.7 rounded to one decimal place. 00:07:40.315 --> 00:07:42.314 What would the units be for our radius? 00:07:42.316 --> 00:07:49.050 Well the area of the sector was in square feet, so our radius should be in feet. 00:07:51.551 --> 00:07:54.576 Next we're going to look at arc length. 00:07:54.578 --> 00:07:58.312 This is the length of an arc or a portion of a circle. 00:07:58.314 --> 00:08:03.082 So I've drawn a circle with radius r. This is the center point. I have central angle 00:08:03.084 --> 00:08:09.089 theta again. And the arc is labelled s, and it's this portion of the circle 00:08:09.091 --> 00:08:13.598 intercepted by this central angle. I'd like to be able to calculate 00:08:13.600 --> 00:08:19.094 the length of that arc knowing just the central angle and the radius. 00:08:19.096 --> 00:08:27.843 I'd like to look again at the fraction of the circle represented by, in this case, the arc. 00:08:27.845 --> 00:08:37.117 So we're going to do the arc length over the circumference, or distance around 00:08:37.119 --> 00:08:45.875 the entire circle, and then we're going to look again at the angle. 00:08:45.877 --> 00:08:53.134 So the central angle for this arc is theta, and an angle that starts here and goes 00:08:53.136 --> 00:08:58.639 all the way around the circle is 2 pi radians. So again, we're going to have that 00:08:58.641 --> 00:09:04.645 stipulation that theta has to be in radians. Now we've written the 00:09:04.647 --> 00:09:10.302 fraction of the circle present two ways. But they're the same fraction of the same circle, 00:09:10.304 --> 00:09:18.100 so they have to be equal. For arclength, we're going to use the letter s. 00:09:18.102 --> 00:09:21.161 Circumference of a circle, do you remember that formula? 00:09:21.163 --> 00:09:40.400 2 pi times the radius of the circle. Let's solve for s. Multiply both sides by 2 pi r. 00:09:40.442 --> 00:09:46.184 Notice that the 2 pi's divide out for us, and we're left with a very simple formula, 00:09:46.186 --> 00:09:57.410 s equals r theta, theta in radians. 00:09:58.949 --> 00:10:04.440 Let's practice using our arc length formula. We'd like to find the exact length of an arc 00:10:04.452 --> 00:10:10.460 of a circle with radius 10 feet intercepted by central angle 330 degrees. 00:10:10.462 --> 00:10:23.719 So we have this nice formula, s equals r theta, but theta has to be in radians. 00:10:23.721 --> 00:10:28.227 We're given the central angle of 330 degrees, so before we do anything else, we need 00:10:28.229 --> 00:10:36.988 to go ahead and change this to radians. There are pi radians in 180 degrees, 00:10:36.990 --> 00:10:47.501 so we're going to multiply by pi over 180. So this one is just 33 pi over 18. 00:10:47.503 --> 00:10:54.000 Oh wait, I can go further. Those both have a common factor of 3, 00:10:54.002 --> 00:11:04.502 so this is really 11 pi over 6. Now we substitute. The radius is 10 feet. Theta is 00:11:04.504 --> 00:11:18.028 11 pi over 6. Think of this as 10 over 1 and reduce your fractions. 00:11:18.030 --> 00:11:23.028 So I'm going to just go ahead and multiply, 10 times 11 is 110 pi over 6, 00:11:23.030 --> 00:11:32.042 and then reduce that. What kind of units would arc length have? 00:11:32.044 --> 00:11:38.545 Well it's just a length, so whatever the unit is used for the radius we'll use for 00:11:38.547 --> 00:11:48.308 our arclength so 10 feet. We wanted an exact value, so we want to leave it in terms 00:11:48.310 --> 00:11:50.560 of pi and an improper fraction. 00:11:52.638 --> 00:11:59.070 Here's another example that uses our arc length formula. Find the central angle 00:11:59.072 --> 00:12:04.317 of a circle with radius 30 inches that intercepts an arc of length 10 inches. 00:12:04.319 --> 00:12:13.070 So we can use our arc length formula, s equals r theta, and in this case, 00:12:13.072 --> 00:12:17.068 we're looking for the central angle. That's what we don't know. 00:12:17.070 --> 00:12:25.070 Our radius is 30 inches, and the arc length is 10 inches. So an arc of length 10 inches, 00:12:25.072 --> 00:12:37.360 this is the s in our formula. r is 30. If we divide both sides by 30, 00:12:37.362 --> 00:12:46.620 we'll have theta by itself already. So theta is 10 over 30 or 1/3. 1/3 what? 00:12:46.622 --> 00:12:50.353 What are the units for our angle theta? 00:12:50.355 --> 00:12:54.865 Right. So theta, in order to use this formula, has to be in radians. 00:12:54.867 --> 00:13:02.617 So when we solve this, we're finding our angle theta in radians. 00:13:02.619 --> 00:13:07.630 If a particular problem asks you for that answer in degrees, then you know how to 00:13:07.632 --> 00:13:12.383 convert from radians to degrees, and you can do that at that time. 00:13:12.394 --> 00:13:17.141 So you need to learn both of these formulas for area of a sector and for arc length. 00:13:17.143 --> 00:13:21.640 Don't forget that your angle, theta, has to be in radians each time that you use them, 00:13:21.642 --> 00:13:27.900 and then practice these examples until you're comfortable with these skills.