1 00:00:00,550 --> 00:00:07,300 In this video, let's discuss faces at Justice and what he says, and then we'll take you to Euler's 2 00:00:07,300 --> 00:00:07,780 formula. 3 00:00:07,990 --> 00:00:11,200 All right, now over here, let's get started with the basics. 4 00:00:11,470 --> 00:00:13,610 I have a cube over here. 5 00:00:13,900 --> 00:00:15,310 These are the faces. 6 00:00:16,340 --> 00:00:19,430 Of the cube, that is, they are just flat surfaces. 7 00:00:20,750 --> 00:00:27,830 All right, now, two faces for Cube meet at the nine segment, and we call that and it's an example 8 00:00:27,830 --> 00:00:28,410 of an end. 9 00:00:28,430 --> 00:00:29,900 Is this part over here? 10 00:00:30,620 --> 00:00:31,160 All right. 11 00:00:31,310 --> 00:00:37,700 And in the case of a cube, three adults meet at a point called the vertex, for example. 12 00:00:37,850 --> 00:00:41,540 You have a vortex over here now in every shape. 13 00:00:41,550 --> 00:00:42,770 It need not be three. 14 00:00:42,920 --> 00:00:48,890 But in the case of a cube and for our understanding purpose in this case, you can see one, two and 15 00:00:48,890 --> 00:00:52,010 three three and just meet at Vertex. 16 00:00:52,430 --> 00:00:52,910 All right. 17 00:00:53,180 --> 00:00:57,980 Now, the plural is vertices and the singular is vertex. 18 00:00:59,110 --> 00:01:06,010 All right, now let's continue analyzing the relation between faces at Justice and what he says now 19 00:01:06,010 --> 00:01:10,350 let's see the relation for these five types of shapes. 20 00:01:10,600 --> 00:01:17,320 You have a cuboid, a triangular pyramid, triangular prism, a pyramid with the square base and a prism 21 00:01:17,320 --> 00:01:18,340 with a square base. 22 00:01:18,400 --> 00:01:24,310 All right, let's get started and we will try to find the faces, what it says and just what each of 23 00:01:24,310 --> 00:01:26,620 these not over here you have a cuboid. 24 00:01:26,920 --> 00:01:28,550 So this is a quote right now. 25 00:01:28,570 --> 00:01:30,370 You can see that it has six faces. 26 00:01:30,520 --> 00:01:31,510 We've already seen that. 27 00:01:31,510 --> 00:01:31,730 Right? 28 00:01:31,960 --> 00:01:32,680 This is one. 29 00:01:32,830 --> 00:01:33,430 This is two. 30 00:01:33,430 --> 00:01:33,970 This is three. 31 00:01:33,970 --> 00:01:36,360 And then you will have three more, which you can see from here. 32 00:01:36,700 --> 00:01:38,160 And how many what is this? 33 00:01:38,200 --> 00:01:42,190 Do you have we have one, two, three, four over here. 34 00:01:42,370 --> 00:01:44,560 And similarly, you will have four over here also. 35 00:01:44,710 --> 00:01:46,570 So that's eight what it says. 36 00:01:46,720 --> 00:01:48,640 And it has 12 edges. 37 00:01:48,640 --> 00:01:48,880 Right. 38 00:01:49,030 --> 00:01:52,510 Your forehead just on the top or just on the bottom. 39 00:01:52,510 --> 00:01:54,310 And then this is one and two. 40 00:01:54,470 --> 00:01:55,680 Similarly one and two. 41 00:01:55,690 --> 00:01:57,190 So four plus four plus four. 42 00:01:57,370 --> 00:01:58,590 That's twelve edges. 43 00:01:58,750 --> 00:02:02,770 So that's six faces, eight vertices and twelve edges. 44 00:02:02,980 --> 00:02:03,390 All right. 45 00:02:03,550 --> 00:02:05,350 Now let's take the same faces. 46 00:02:05,350 --> 00:02:09,100 Plus what it says that six plus eight, it gives you 14. 47 00:02:09,100 --> 00:02:09,450 Right. 48 00:02:09,670 --> 00:02:13,330 Similarly, if you take it just plus two, that's twelve plus two. 49 00:02:13,510 --> 00:02:15,760 That's also equal to fourteen. 50 00:02:16,390 --> 00:02:17,380 Interesting, right? 51 00:02:17,590 --> 00:02:19,550 These two values are coming as equal. 52 00:02:19,850 --> 00:02:21,970 Now, let's try a triangular pyramid. 53 00:02:22,240 --> 00:02:24,670 You must have seen the pyramids in Egypt, right. 54 00:02:24,880 --> 00:02:30,190 So this type of shape is called the triangular pyramid, and it's called a triangular pyramid because 55 00:02:30,310 --> 00:02:32,880 the base is of a shape of a triangle. 56 00:02:32,900 --> 00:02:33,250 Right. 57 00:02:33,520 --> 00:02:36,180 These two sides will be joined right now. 58 00:02:36,190 --> 00:02:38,170 How many faces does it have? 59 00:02:38,290 --> 00:02:39,910 It has one, two and three. 60 00:02:40,000 --> 00:02:40,900 And the bottom. 61 00:02:40,900 --> 00:02:42,000 That's four faces. 62 00:02:42,020 --> 00:02:42,330 Right. 63 00:02:42,580 --> 00:02:44,470 And how many bodies does it have? 64 00:02:44,650 --> 00:02:45,480 It has three. 65 00:02:45,490 --> 00:02:47,800 What is at the bottom and one over here. 66 00:02:48,040 --> 00:02:49,300 That's four vertices. 67 00:02:49,480 --> 00:02:53,680 And you can see that it has six edges, which are the six edges. 68 00:02:53,860 --> 00:02:57,160 You have three at the bottom and then one, two and three over here. 69 00:02:57,340 --> 00:02:59,170 The three plus three equals six. 70 00:02:59,350 --> 00:02:59,800 All right. 71 00:02:59,980 --> 00:03:02,110 Now let's dig phases plus what he says. 72 00:03:02,260 --> 00:03:04,600 That's four plus four, which gives you eight. 73 00:03:04,810 --> 00:03:05,730 And what about it? 74 00:03:05,740 --> 00:03:06,580 Just plus two? 75 00:03:06,760 --> 00:03:08,900 That six plus two, which gives you eight. 76 00:03:09,280 --> 00:03:09,730 All right. 77 00:03:09,730 --> 00:03:10,660 They're coming equal. 78 00:03:10,840 --> 00:03:12,010 That's also interesting. 79 00:03:12,220 --> 00:03:14,880 Now, let's take a triangular prism, OK? 80 00:03:14,890 --> 00:03:16,810 This is a triangular prism over here. 81 00:03:16,810 --> 00:03:20,980 You have a triangular shape and the bottom also has a triangular base. 82 00:03:20,980 --> 00:03:21,280 Right. 83 00:03:21,520 --> 00:03:23,560 And these sides are rectangles. 84 00:03:23,740 --> 00:03:25,920 This type of a shape is called a prism. 85 00:03:25,930 --> 00:03:31,810 And because the top and bottom base are in the form of a triangle, it's called a triangular prism. 86 00:03:32,020 --> 00:03:32,460 All right. 87 00:03:32,650 --> 00:03:35,600 Now, let's try to find a number of faces over here. 88 00:03:35,830 --> 00:03:38,230 You have one, two, three, four and five. 89 00:03:38,230 --> 00:03:38,510 Right. 90 00:03:38,800 --> 00:03:42,030 Two at the bottom and top and then three in the sides. 91 00:03:42,030 --> 00:03:43,750 So two plus three is equal to five. 92 00:03:43,990 --> 00:03:49,390 So you have five faces and you can see that you have three vertices over here and three vertices over 93 00:03:49,390 --> 00:03:49,690 here. 94 00:03:49,900 --> 00:03:51,190 That's six what he says. 95 00:03:51,340 --> 00:03:53,140 And it has nine edges. 96 00:03:53,140 --> 00:03:53,500 Right. 97 00:03:53,650 --> 00:03:57,930 You have three edges over here, three over here, and then one, two and three over here. 98 00:03:57,940 --> 00:04:00,870 So three plus three plus three, that's nine edges. 99 00:04:00,880 --> 00:04:01,290 All right. 100 00:04:01,450 --> 00:04:03,550 Now let's take Frasers plus what he says. 101 00:04:03,700 --> 00:04:05,800 That's five plus six, which gives you eleven. 102 00:04:06,160 --> 00:04:08,020 And let's take it just plus two. 103 00:04:08,020 --> 00:04:10,960 That's nine plus two, which gives you also eleven. 104 00:04:11,290 --> 00:04:11,730 All right. 105 00:04:11,860 --> 00:04:14,970 Now, let's go ahead with the pyramid, with the square base. 106 00:04:15,160 --> 00:04:19,450 Now, imagine you had this same shape over here, but you have a square base. 107 00:04:20,230 --> 00:04:20,710 All right. 108 00:04:20,890 --> 00:04:22,630 Now, how many faces will it have? 109 00:04:22,810 --> 00:04:24,520 It will have four plus one. 110 00:04:24,520 --> 00:04:28,690 That's five faces and four vertices at the bottom and one at the top. 111 00:04:28,690 --> 00:04:29,890 That's five vertices. 112 00:04:30,100 --> 00:04:31,780 And it'll have eight edges. 113 00:04:31,780 --> 00:04:34,270 Right, or at the bottom and four over here. 114 00:04:34,270 --> 00:04:35,530 One, two, three and four. 115 00:04:35,800 --> 00:04:37,500 But I love eight just right. 116 00:04:37,630 --> 00:04:39,910 So phases plus what it says gives you ten. 117 00:04:40,480 --> 00:04:40,960 Right. 118 00:04:40,960 --> 00:04:42,220 And it just plus two. 119 00:04:42,220 --> 00:04:43,270 That's eight plus two. 120 00:04:43,270 --> 00:04:44,770 That also gives you ten. 121 00:04:45,100 --> 00:04:45,570 All right. 122 00:04:45,760 --> 00:04:51,850 Now let's check a with the square base that look like this and you can see that this is the same as 123 00:04:51,850 --> 00:04:52,500 a cuboid. 124 00:04:52,510 --> 00:04:53,340 Both are the same. 125 00:04:53,560 --> 00:04:54,010 All right. 126 00:04:54,220 --> 00:04:56,740 Now over here, this is a prism, right? 127 00:04:56,740 --> 00:04:58,990 And it has a base in the shape of a square. 128 00:04:59,170 --> 00:05:01,400 That's why it's called a prism with a square base. 129 00:05:01,660 --> 00:05:03,430 Now, how many faces does it have? 130 00:05:03,550 --> 00:05:04,750 It has six faces. 131 00:05:04,910 --> 00:05:05,380 How many? 132 00:05:05,380 --> 00:05:08,800 What is says one, two, three, four and four at the bottom. 133 00:05:09,010 --> 00:05:10,780 That's eight what it says. 134 00:05:10,780 --> 00:05:14,500 And it has 12 edges or over here or at the bottom. 135 00:05:14,500 --> 00:05:16,900 And then one, two, three and one more over here. 136 00:05:17,050 --> 00:05:17,620 That's four. 137 00:05:17,710 --> 00:05:19,330 So four plus four plus four. 138 00:05:19,450 --> 00:05:21,460 That's four twelve edges just now. 139 00:05:21,460 --> 00:05:25,750 Let's take faces plus what it says that six plus eight which is equal to fourteen. 140 00:05:25,990 --> 00:05:26,800 And what about it. 141 00:05:26,800 --> 00:05:27,580 Just plus two. 142 00:05:27,760 --> 00:05:30,680 That's twelve plus two which is also equal to fourteen. 143 00:05:30,970 --> 00:05:35,350 Now this is a beautiful property which we have seen over here. 144 00:05:35,500 --> 00:05:42,820 That is the number of faces plus the number of what is says is equal to the number of it just plus two. 145 00:05:43,000 --> 00:05:46,230 And this is what we call the Euler's formula. 146 00:05:46,510 --> 00:05:47,020 All right. 147 00:05:47,140 --> 00:05:49,960 And this is true for any Hedren. 148 00:05:51,120 --> 00:05:57,090 All right, now, let's take an example of a question to analyze the utility or the usefulness of this 149 00:05:57,090 --> 00:05:57,660 formula. 150 00:05:57,960 --> 00:05:58,410 All right. 151 00:05:58,410 --> 00:05:59,700 Over here we have a question. 152 00:05:59,850 --> 00:06:02,220 A politician has seven faces and 10. 153 00:06:02,220 --> 00:06:02,850 What is this? 154 00:06:03,090 --> 00:06:06,080 How many it just does the police have. 155 00:06:06,390 --> 00:06:08,170 All right, let's get started. 156 00:06:08,490 --> 00:06:11,550 Now, we know that the faces is equal to seven. 157 00:06:11,710 --> 00:06:13,290 The what it says is equal to ten. 158 00:06:13,290 --> 00:06:15,480 And we need to find the number of it just. 159 00:06:16,340 --> 00:06:22,490 Now faces plus what he says is equal to it, just plus two as per the Oilor formula, therefore we can 160 00:06:22,490 --> 00:06:25,880 solve that the number of adults will be equal to 15. 161 00:06:26,120 --> 00:06:27,530 Now, let's do one more question. 162 00:06:27,770 --> 00:06:31,700 The Euler's formula is true for all three dimensional shapes. 163 00:06:32,030 --> 00:06:33,170 Now, is that true? 164 00:06:33,410 --> 00:06:34,400 No, it's not true. 165 00:06:34,400 --> 00:06:34,670 Right. 166 00:06:34,880 --> 00:06:37,550 We have seen that it's true only for Poly Heidrun, right? 167 00:06:37,610 --> 00:06:39,740 Not all three dimensional shapes. 168 00:06:40,090 --> 00:06:41,320 Therefore, this is false. 169 00:06:41,540 --> 00:06:47,310 And this formula works for all politicians except once having holes through it. 170 00:06:47,330 --> 00:06:50,240 Now, don't worry, you will not be getting these type of questions. 171 00:06:50,420 --> 00:06:55,990 But only headrests that have a hole through it are not simple polygons. 172 00:06:56,310 --> 00:06:59,450 OK, so they are called non simple politicians.