1 00:00:00,001 --> 00:00:07,991 [music] We know what the area of a rectangle is. 2 00:00:07,992 --> 00:00:16,099 Well, if you've got a rectangle, say, of width a and height b, what should the area 3 00:00:16,099 --> 00:00:22,890 of this rectangle be? The area of this rectangle is a times b. 4 00:00:22,890 --> 00:00:31,281 So what's the area of a triangle? Well, if we've got a, triangle say of 5 00:00:31,281 --> 00:00:40,462 height b and width a, right. The area of this triangle is 1 half ab. 6 00:00:40,463 --> 00:00:45,607 But why? Why is that the area of a triangle? 7 00:00:45,608 --> 00:00:49,680 Well this is really what the area of the triangle has to be in order to be 8 00:00:49,680 --> 00:00:52,617 consistent with the area of the whole rectangle. 9 00:00:52,618 --> 00:00:57,465 I mean, look, I could take two of these triangles. 10 00:00:57,466 --> 00:01:02,891 Right here's two triangles, both have width a and height b. 11 00:01:02,891 --> 00:01:09,479 But if I push those 2 triangles together, then I've got the a by b rectangle. 12 00:01:09,479 --> 00:01:16,220 So the area of these 2 triangles together has to be the area of the rectangle a 13 00:01:16,220 --> 00:01:20,071 times b. But each of these triangles has half the 14 00:01:20,071 --> 00:01:24,472 area of the rectangle. So the area of just the triangle should be 15 00:01:24,472 --> 00:01:29,484 half the area of the rectangle. Another way to say this is just that, you 16 00:01:29,484 --> 00:01:33,358 know, areas sort of add. You know, area of some figure plus area of 17 00:01:33,358 --> 00:01:37,538 some other figure. Is equal to the area of those two things 18 00:01:37,538 --> 00:01:38,793 together. Right? 19 00:01:38,793 --> 00:01:44,240 In this particular case, the area of this triangle plus the area of this triangle, 20 00:01:44,240 --> 00:01:49,137 is the area of these two things together, which is the area of a rectangle. 21 00:01:49,138 --> 00:01:54,296 You can play this kind of game to relate the areas of different figures that 22 00:01:54,296 --> 00:02:00,031 supposedly have the same area. For example, I could start with a 23 00:02:00,031 --> 00:02:07,551 rectangle of width 2 and height 1. Or, I could start with a square whose side 24 00:02:07,551 --> 00:02:13,868 length is the square root of 2. Now, both of these figures have the same 25 00:02:13,868 --> 00:02:18,128 area, right? The area in both of these figures is 2. 26 00:02:18,128 --> 00:02:23,116 And you might ask, well, you know, if the whole deal with areas is just so that they 27 00:02:23,116 --> 00:02:27,604 sort of cut up and add appropriately. Is it possible for me to cut up this 28 00:02:27,604 --> 00:02:32,834 figure in order to get this figure? And, and yeah, it is in fact possible. 29 00:02:32,834 --> 00:02:38,684 Right? If I cut along the diagonals here, these 4 30 00:02:38,684 --> 00:02:45,664 pieces can be rearranged into these 4 pieces here. 31 00:02:45,664 --> 00:02:50,393 So, and very visibly, right, if you believe that, you know, sort of areas add 32 00:02:50,393 --> 00:02:53,864 in this way that I'm allowed to rearrange pieces. 33 00:02:53,865 --> 00:02:59,410 Then you might believe that these 2 things have the same area, regardless of whether 34 00:02:59,410 --> 00:03:04,376 you do the calculation or not, right? These 2 pieces have the same area because 35 00:03:04,376 --> 00:03:07,707 I can take this thing and cut it up and get this thing. 36 00:03:07,708 --> 00:03:10,670 Well here's a challenge for you right along these lines. 37 00:03:10,670 --> 00:03:23,164 For instance can you take a 3x1 rectangle, and a square of side length root 3. 38 00:03:23,164 --> 00:03:26,410 And visibly show that these things have the same area. 39 00:03:26,410 --> 00:03:29,806 I mean yes. The calculation, right, is telling me that 40 00:03:29,806 --> 00:03:33,832 they both have the same area of 3, 1 times 3 is the same as the square root of 3 41 00:03:33,832 --> 00:03:37,450 times the square root of 3. But what I'm looking for is some sort of 42 00:03:37,450 --> 00:03:41,740 geometric proof where you cut up this thing and rearrange those pieces to build 43 00:03:41,740 --> 00:03:44,230 this thing, right. That's your challenge. 44 00:03:44,230 --> 00:03:47,232 So this is really fascinating, right. What is area? 45 00:03:47,232 --> 00:03:53,007 Area is a number, but that number captures this geometric idea that if two things 46 00:03:53,007 --> 00:03:58,683 have the same number, the same area, then you can take the one thing and chop it up 47 00:03:58,683 --> 00:04:03,500 into pieces to get the other thing. And yet that perspective is totally 48 00:04:03,500 --> 00:04:08,516 destroyed as soon as we start thinking about something with a curved boundary, 49 00:04:08,516 --> 00:04:12,617 like a circle. So here's a unit circle by which I mean 50 00:04:12,617 --> 00:04:17,228 the radius has length 1 unit. In the area of the unit circle is pi 51 00:04:17,228 --> 00:04:21,310 square units. I mean yes we know what the formula for 52 00:04:21,310 --> 00:04:27,421 the area of a circle says right, it says that the area of that unit circle Is pi 53 00:04:27,421 --> 00:04:30,400 square units. But what does that mean? 54 00:04:30,400 --> 00:04:34,521 What does it mean to say the circle's area is pi square units? 55 00:04:34,522 --> 00:04:40,656 Well I'll tell you what it doesn't mean. It doesn't mean that I can start with a 56 00:04:40,656 --> 00:04:45,932 rectangle of area pi square units. So say a rectangle of height 1. 57 00:04:45,932 --> 00:04:49,918 And with pi, right? To say that the area of this circle is 58 00:04:49,918 --> 00:04:55,126 equal to pi does not mean that if I just start with a rectangle with that same 59 00:04:55,126 --> 00:05:00,760 area, that I can necessarily make some finite number of cuts to this rectangle. 60 00:05:00,760 --> 00:05:04,353 And take those pieces and rearrange them to form this disc. 61 00:05:04,354 --> 00:05:08,124 All right. There's really two competing concepts 62 00:05:08,124 --> 00:05:11,090 here. You've already got an idea of area, you 63 00:05:11,090 --> 00:05:13,466 know? Because you've seen area before. 64 00:05:13,467 --> 00:05:18,780 And this idea of area is really based on this idea that two things have the same 65 00:05:18,780 --> 00:05:24,009 area if you can cut the one thing up and rearrange the pieces to make the other 66 00:05:24,009 --> 00:05:27,790 thing. But as soon as you're dealing with objects 67 00:05:27,790 --> 00:05:32,600 with curved boundaries, say, you've suddenly got a problem, because it's not 68 00:05:32,600 --> 00:05:37,188 going to be possible for you to take this object and just chop it up and produce 69 00:05:37,188 --> 00:05:43,772 this nice smoothly curved object. So, somehow you've gotta refine what area 70 00:05:43,772 --> 00:05:47,780 even means. When I say that the area of a circle is pi 71 00:05:47,780 --> 00:05:52,368 square units, I'm secretly speaking in the language of limits. 72 00:05:52,368 --> 00:05:58,167 So, when I say that this circle has area pi, I mean that if I say cut it up into 73 00:05:58,167 --> 00:06:03,560 small enough little rectangles. The area of these rectangles, which 74 00:06:03,560 --> 00:06:07,572 actually makes sense, because I know how to calculate the areas of rectangles, by 75 00:06:07,572 --> 00:06:10,041 multiplying their widths times their heights. 76 00:06:10,042 --> 00:06:15,345 So, if I add up the areas of these rectangles, I get an answer that's as 77 00:06:15,345 --> 00:06:20,960 close as I'd like to the number pi. And I can get the answer as close as I 78 00:06:20,960 --> 00:06:24,507 want, provided I make these rectangles thin enough. 79 00:06:24,508 --> 00:06:29,850 And it's only in that sense that it, it even means anything to say, that the area 80 00:06:29,850 --> 00:06:33,139 of the circle is pi. It's in the sense of that limit. 81 00:06:33,140 --> 00:06:38,536 So even defining area requires talking about limits, which I think is remarkable. 82 00:06:38,536 --> 00:06:42,184 I mean we learn about area in elementary school, right? 83 00:06:42,184 --> 00:06:45,388 We never talk about limits in elementary school. 84 00:06:45,388 --> 00:06:49,810 And yet somehow there's a real depth, a real subtlety to even this concept of 85 00:06:49,810 --> 00:06:53,226 area, you know? And I think that's really exciting. 86 00:06:53,226 --> 00:06:57,900 Right? It suggests that there's something 87 00:06:57,900 --> 00:07:07,715 interesting yet to be learned even about concepts that we think we understand 88 00:07:07,715 --> 00:07:09,433 really well.