WEBVTT 00:00:00.830 --> 00:00:03.240 So right here, we have a four-sided figure, 00:00:03.240 --> 00:00:06.530 or a quadrilateral, where two of the sides 00:00:06.530 --> 00:00:08.520 are parallel to each other. 00:00:08.520 --> 00:00:10.636 And so this, by definition, is a trapezoid. 00:00:14.530 --> 00:00:16.570 And what we want to do is, given the dimensions 00:00:16.570 --> 00:00:20.630 that they've given us, what is the area of this trapezoid. 00:00:20.630 --> 00:00:22.660 So let's just think through it. 00:00:22.660 --> 00:00:26.250 So what would we get if we multiplied this long base 00:00:26.250 --> 00:00:28.670 6 times the height 3? 00:00:28.670 --> 00:00:33.750 So what do we get if we multiply 6 times 3? 00:00:33.750 --> 00:00:35.900 Well, that would be the area of a rectangle that 00:00:35.900 --> 00:00:39.790 is 6 units wide and 3 units high. 00:00:39.790 --> 00:00:42.530 So that would give us the area of a figure that 00:00:42.530 --> 00:00:44.980 looked like-- let me do it in this pink color. 00:00:44.980 --> 00:00:49.940 The area of a figure that looked like this would be 6 times 3. 00:00:49.940 --> 00:00:53.790 So it would give us this entire area right over there. 00:00:53.790 --> 00:00:55.760 Now, the trapezoid is clearly less than that, 00:00:55.760 --> 00:00:58.770 but let's just go with the thought experiment. 00:00:58.770 --> 00:01:04.980 Now, what would happen if we went with 2 times 3? 00:01:04.980 --> 00:01:07.910 Well, now we'd be finding the area of a rectangle that 00:01:07.910 --> 00:01:10.260 has a width of 2 and a height of 3. 00:01:10.260 --> 00:01:14.810 So you could imagine that being this rectangle right over here. 00:01:14.810 --> 00:01:18.240 So that is this rectangle right over here. 00:01:18.240 --> 00:01:22.130 So that's the 2 times 3 rectangle. 00:01:22.130 --> 00:01:26.160 Now, it looks like the area of the trapezoid 00:01:26.160 --> 00:01:28.910 should be in between these two numbers. 00:01:28.910 --> 00:01:32.490 Maybe it should be exactly halfway in between, 00:01:32.490 --> 00:01:36.050 because when you look at the area difference between the two 00:01:36.050 --> 00:01:39.240 rectangles-- and let me color that in. 00:01:39.240 --> 00:01:43.030 So this is the area difference on the left-hand side. 00:01:43.030 --> 00:01:48.980 And this is the area difference on the right-hand side. 00:01:48.980 --> 00:01:51.090 If we focus on the trapezoid, you 00:01:51.090 --> 00:01:56.480 see that if we start with the yellow, the smaller rectangle, 00:01:56.480 --> 00:01:59.610 it reclaims half of the area, half 00:01:59.610 --> 00:02:03.030 of the difference between the smaller rectangle 00:02:03.030 --> 00:02:05.240 and the larger one on the left-hand side. 00:02:05.240 --> 00:02:07.920 It gets exactly half of it on the left-hand side. 00:02:07.920 --> 00:02:10.050 And it gets half the difference between the smaller 00:02:10.050 --> 00:02:12.290 and the larger on the right-hand side. 00:02:12.290 --> 00:02:17.260 So it completely makes sense that the area 00:02:17.260 --> 00:02:20.420 of the trapezoid, this entire area right over here, 00:02:20.420 --> 00:02:22.310 should really just be the average. 00:02:22.310 --> 00:02:25.420 It should exactly be halfway between the areas 00:02:25.420 --> 00:02:28.172 of the smaller rectangle and the larger rectangle. 00:02:28.172 --> 00:02:30.130 So let's take the average of those two numbers. 00:02:30.130 --> 00:02:38.160 It's going to be 6 times 3 plus 2 times 3, all of that over 2. 00:02:38.160 --> 00:02:40.230 So when you think about an area of a trapezoid, 00:02:40.230 --> 00:02:44.940 you look at the two bases, the long base and the short base. 00:02:47.840 --> 00:02:50.410 Multiply each of those times the height, and then you 00:02:50.410 --> 00:02:51.720 could take the average of them. 00:02:51.720 --> 00:02:53.680 Or you could also think of it as this 00:02:53.680 --> 00:02:57.440 is the same thing as 6 plus 2. 00:02:57.440 --> 00:02:59.490 And I'm just factoring out a 3 here. 00:02:59.490 --> 00:03:12.760 6 plus 2 times 3, and then all of that over 2, 00:03:12.760 --> 00:03:14.274 which is the same thing as-- and I'm 00:03:14.274 --> 00:03:15.690 just writing it in different ways. 00:03:15.690 --> 00:03:17.690 These are all different ways to think about it-- 00:03:17.690 --> 00:03:25.450 6 plus 2 over 2, and then that times 3. 00:03:25.450 --> 00:03:27.820 So you could view it as the average 00:03:27.820 --> 00:03:30.560 of the smaller and larger rectangle. 00:03:30.560 --> 00:03:32.790 So you multiply each of the bases times the height 00:03:32.790 --> 00:03:34.180 and then take the average. 00:03:34.180 --> 00:03:37.540 You could view it as-- well, let's just add up the two base 00:03:37.540 --> 00:03:41.360 lengths, multiply that times the height, and then divide by 2. 00:03:41.360 --> 00:03:43.710 Or you could say, hey, let's take the average of the two 00:03:43.710 --> 00:03:46.481 base lengths and multiply that by 3. 00:03:46.481 --> 00:03:48.230 And that gives you another interesting way 00:03:48.230 --> 00:03:48.980 to think about it. 00:03:48.980 --> 00:03:52.850 If you take the average of these two lengths, 6 plus 2 over 2 00:03:52.850 --> 00:03:54.660 is 4. 00:03:54.660 --> 00:03:57.770 So that would be a width that looks something 00:03:57.770 --> 00:03:59.690 like-- let me do this in orange. 00:03:59.690 --> 00:04:03.080 A width of 4 would look something like this. 00:04:03.080 --> 00:04:05.000 A width of 4 would look something like that, 00:04:05.000 --> 00:04:07.050 and you're multiplying that times the height. 00:04:07.050 --> 00:04:11.440 Well, that would be a rectangle like this that is exactly 00:04:11.440 --> 00:04:14.190 halfway in between the areas of the small 00:04:14.190 --> 00:04:16.089 and the large rectangle. 00:04:16.089 --> 00:04:18.420 So these are all equivalent statements. 00:04:18.420 --> 00:04:20.010 Now let's actually just calculate it. 00:04:20.010 --> 00:04:21.176 So we could do any of these. 00:04:21.176 --> 00:04:24.120 6 times 3 is 18. 00:04:24.120 --> 00:04:28.630 This is 18 plus 6, over 2. 00:04:28.630 --> 00:04:31.501 That is 24/2, or 12. 00:04:31.501 --> 00:04:32.750 You could also do it this way. 00:04:32.750 --> 00:04:38.090 6 plus 2 is 8, times 3 is 24, divided by 2 is 12. 00:04:38.090 --> 00:04:42.430 6 plus 2 divided by 2 is 4, times 3 is 12. 00:04:42.430 --> 00:04:47.600 Either way, the area of this trapezoid is 12 square units.