High students in this video, let's analyze the case where you repeatedly inscribe a circle and a square
within each other.
All right.
Now we will be analyzing the area of the circles and the squares.
All right.
Over here you have a square.
Let the side of the square be.
Now you can see that the area of this square is a square.
All right.
Now let's inscribe a circle inside this square that would look like this.
Now you can see that the diameter of this circle is equal to the side of the square.
Right.
Therefore, two are is equal to it or is equal to two.
Right.
Therefore, the area of this circle will be by in two by two.
The whole square that's equal to buy by four into a square.
Now imagine that we inscribe a square inside this circle, but that would look like this, right?
OK, now let's analyze this.
You can see that if the side of this square is beat, then this diagonal over here will be equal to
the site of the initial square.
Right.
Why is that so?
Because you can see that over here.
You have the diameter of this circle equal to the diagonal of this square, right, and initially we
had seen that the diameter of that circle is equal to it.
Therefore, if this is B and this is B, then this diagonal is route to B, and that would be equal
to A, right.
So let's try that over here to be is equal to it.
So I can say that B is equal to a Beirut two.
All right.
Now, what would be the area of this second square over here that would be being to be which is equal
to a rotu into a to that's a square by two.
All right.
Now, let me clear this space now.
If you were to inscribe one more circle inside this square over here, can you tell me what would be
the area that would follow the same pattern?
Right over here you have a square and you have to multiply that by by four to get the area of the circle.
Similarly, over here, you have a square by two.
So you need to multiply that with by by four and the area of the circle that you inscribed inside this
square.
Right.
This one over here, that area would be by by four in a square by two.
And that is equal to buy by eight into a square.
All right.
Now, let's keep this aside.
Now you can see that there is a pattern over here, right?
The ratio remains constant.
That is, the ratio between this square area and this circle area remains constant.
Now, let's see the squares over here.
And even the ratio between this square and this squares area is also constant.
Right.
All right.
Now, let's only take the squares.
You will see that the area is over here is square.
And then over here, you got a square by two and the next one would be square by four.
And it will go like that right now if you take only circles.
Now, let's take this area over here.
That is five by four into a square.
As see, then the areas of the Sakalys would be you have C then you have see by two, then you see by
four and it will go on like that.
Now, if you write these two together, the progression would be like this.
You have a square then by by four into a square.
Then you have square by two, then by by eight into a square.
And it goes on like that.
Now you can note that these two progressions are geometric progressions and you can see that the common
ratio is equal to one by two.
Right.
Because this stone by this term is equal to one by two.
Now you can find the sum of these areas up to infinity.
If you are asked to find the sum of all the possible squares that can be inscribed like this, then
the answer would be the first term divided by one minus the common ratio.
Right now, you need to refer to geometric progressions to understand why this is so, and this would
be equal to a square divided by one minus one by two, that's equal to two is square.
That would be the sum of all these areas up to infinity.
And you can do the same thing over here.
Also, it would be C by one, minus one by two, which is equal to to see in the case of circles.
That is if you take the sum of all these circles.
Right.
This one over here, then this one over here, and you keep doing that up to infinity, then the sum
of all those areas will be equal to 2C.