1
00:00:00,360 --> 00:00:07,740
High students in this video, let's analyze the case where you repeatedly inscribe a circle and a square

2
00:00:07,890 --> 00:00:09,100
within each other.

3
00:00:09,360 --> 00:00:09,800
All right.

4
00:00:09,930 --> 00:00:14,370
Now we will be analyzing the area of the circles and the squares.

5
00:00:14,640 --> 00:00:15,060
All right.

6
00:00:15,240 --> 00:00:16,640
Over here you have a square.

7
00:00:16,960 --> 00:00:19,150
Let the side of the square be.

8
00:00:19,440 --> 00:00:22,900
Now you can see that the area of this square is a square.

9
00:00:23,340 --> 00:00:23,820
All right.

10
00:00:24,030 --> 00:00:28,600
Now let's inscribe a circle inside this square that would look like this.

11
00:00:28,830 --> 00:00:34,770
Now you can see that the diameter of this circle is equal to the side of the square.

12
00:00:34,830 --> 00:00:35,150
Right.

13
00:00:35,380 --> 00:00:39,720
Therefore, two are is equal to it or is equal to two.

14
00:00:39,720 --> 00:00:40,050
Right.

15
00:00:40,300 --> 00:00:44,580
Therefore, the area of this circle will be by in two by two.

16
00:00:44,580 --> 00:00:48,930
The whole square that's equal to buy by four into a square.

17
00:00:49,260 --> 00:00:55,540
Now imagine that we inscribe a square inside this circle, but that would look like this, right?

18
00:00:56,030 --> 00:00:58,290
OK, now let's analyze this.

19
00:00:58,480 --> 00:01:06,360
You can see that if the side of this square is beat, then this diagonal over here will be equal to

20
00:01:06,450 --> 00:01:08,250
the site of the initial square.

21
00:01:08,340 --> 00:01:08,730
Right.

22
00:01:08,880 --> 00:01:09,850
Why is that so?

23
00:01:10,050 --> 00:01:11,940
Because you can see that over here.

24
00:01:13,110 --> 00:01:21,090
You have the diameter of this circle equal to the diagonal of this square, right, and initially we

25
00:01:21,090 --> 00:01:24,820
had seen that the diameter of that circle is equal to it.

26
00:01:25,080 --> 00:01:32,190
Therefore, if this is B and this is B, then this diagonal is route to B, and that would be equal

27
00:01:32,190 --> 00:01:33,260
to A, right.

28
00:01:33,390 --> 00:01:36,620
So let's try that over here to be is equal to it.

29
00:01:36,810 --> 00:01:39,930
So I can say that B is equal to a Beirut two.

30
00:01:40,250 --> 00:01:40,700
All right.

31
00:01:40,830 --> 00:01:47,010
Now, what would be the area of this second square over here that would be being to be which is equal

32
00:01:47,010 --> 00:01:50,970
to a rotu into a to that's a square by two.

33
00:01:51,330 --> 00:01:51,840
All right.

34
00:01:52,120 --> 00:01:54,510
Now, let me clear this space now.

35
00:01:56,950 --> 00:02:03,100
If you were to inscribe one more circle inside this square over here, can you tell me what would be

36
00:02:03,100 --> 00:02:05,470
the area that would follow the same pattern?

37
00:02:05,470 --> 00:02:12,540
Right over here you have a square and you have to multiply that by by four to get the area of the circle.

38
00:02:12,760 --> 00:02:15,450
Similarly, over here, you have a square by two.

39
00:02:15,670 --> 00:02:22,780
So you need to multiply that with by by four and the area of the circle that you inscribed inside this

40
00:02:22,780 --> 00:02:23,260
square.

41
00:02:23,860 --> 00:02:24,250
Right.

42
00:02:24,610 --> 00:02:30,310
This one over here, that area would be by by four in a square by two.

43
00:02:30,580 --> 00:02:35,440
And that is equal to buy by eight into a square.

44
00:02:35,800 --> 00:02:36,250
All right.

45
00:02:36,250 --> 00:02:37,670
Now, let's keep this aside.

46
00:02:38,080 --> 00:02:40,870
Now you can see that there is a pattern over here, right?

47
00:02:41,050 --> 00:02:43,240
The ratio remains constant.

48
00:02:43,400 --> 00:02:49,540
That is, the ratio between this square area and this circle area remains constant.

49
00:02:49,750 --> 00:02:51,510
Now, let's see the squares over here.

50
00:02:51,700 --> 00:02:56,440
And even the ratio between this square and this squares area is also constant.

51
00:02:56,440 --> 00:02:56,770
Right.

52
00:02:57,040 --> 00:02:57,390
All right.

53
00:02:57,580 --> 00:02:59,650
Now, let's only take the squares.

54
00:02:59,830 --> 00:03:02,760
You will see that the area is over here is square.

55
00:03:03,040 --> 00:03:07,870
And then over here, you got a square by two and the next one would be square by four.

56
00:03:08,020 --> 00:03:11,770
And it will go like that right now if you take only circles.

57
00:03:12,340 --> 00:03:15,100
Now, let's take this area over here.

58
00:03:15,190 --> 00:03:17,590
That is five by four into a square.

59
00:03:17,590 --> 00:03:25,030
As see, then the areas of the Sakalys would be you have C then you have see by two, then you see by

60
00:03:25,030 --> 00:03:27,120
four and it will go on like that.

61
00:03:27,370 --> 00:03:31,630
Now, if you write these two together, the progression would be like this.

62
00:03:31,810 --> 00:03:34,820
You have a square then by by four into a square.

63
00:03:35,020 --> 00:03:38,670
Then you have square by two, then by by eight into a square.

64
00:03:38,800 --> 00:03:40,260
And it goes on like that.

65
00:03:40,510 --> 00:03:46,480
Now you can note that these two progressions are geometric progressions and you can see that the common

66
00:03:46,480 --> 00:03:48,400
ratio is equal to one by two.

67
00:03:48,400 --> 00:03:48,760
Right.

68
00:03:49,030 --> 00:03:52,390
Because this stone by this term is equal to one by two.

69
00:03:52,660 --> 00:03:57,010
Now you can find the sum of these areas up to infinity.

70
00:03:57,190 --> 00:04:03,190
If you are asked to find the sum of all the possible squares that can be inscribed like this, then

71
00:04:03,190 --> 00:04:09,160
the answer would be the first term divided by one minus the common ratio.

72
00:04:11,860 --> 00:04:17,860
Right now, you need to refer to geometric progressions to understand why this is so, and this would

73
00:04:17,860 --> 00:04:24,480
be equal to a square divided by one minus one by two, that's equal to two is square.

74
00:04:24,880 --> 00:04:28,710
That would be the sum of all these areas up to infinity.

75
00:04:28,930 --> 00:04:30,660
And you can do the same thing over here.

76
00:04:30,670 --> 00:04:37,680
Also, it would be C by one, minus one by two, which is equal to to see in the case of circles.

77
00:04:37,690 --> 00:04:40,650
That is if you take the sum of all these circles.

78
00:04:40,660 --> 00:04:40,920
Right.

79
00:04:41,110 --> 00:04:47,290
This one over here, then this one over here, and you keep doing that up to infinity, then the sum

80
00:04:47,290 --> 00:04:50,160
of all those areas will be equal to 2C.