How best to fill a circle with different shapes?!

I was wondering how can I fill a circle with a square.  But, are you wondering, why am I bothered with this? Well, do you know the three “classical problems” in geometry???

The “classical problems” are

• Doubling the cube (i.e., constructing a cube whose volume is twice that of a given cube)
• Trisecting the angle, and 
• Squaring the circle.

Now, are you wondering how is that related to my problem? Well, I thought for a while on the third problem (i.e., squaring the circle), and wondered if I can convert the area of the circle to a square (inscribed in it) then would I find a way to square the circle?

Utter NON-SENSE?!

Yes, I know, I don’t make any sense, don’t I? Well, forget whatever I said and come let us first investigate how to fill a circle with a square…

Too much?

Lets look at it, one by one:

Let r be a radius of a circle and let s be a side of a square inscribed in the circle. So the question is after we draw the square, how much area of the circle will be unoccupied?

Here are the calculations… 

Trivial?

How would the rectangles fill the circle?

A rectangle would have an area always(?) lesser than a square! Interesting, isn’t it?!

What about a triangle inscribed in a circle?

How about a Hexagon?

So, Triangle (3 sides), to square (4 sides), pentagon (5 sides), hexagon (6 sides), and so on…

As the number of sides increases, the unoccupied space reduces from 59% to 36% to X to 17% and so on…

Only a polygon with n (a very large number of) sides could fill the area of the circle or a circle with the same radius! 😁

Coming to the Three classical problems, why are these problems still a problem even in this modern time?

Well, Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with a compass and straightedge. So without a ruler (scale), it is a challenge to square a circle!!!

For one to square a circle, the diagonal of the square should be greater than the diameter of the circle!

Trivial isn’t it? Well, I have only reached that far… 😀

The main challenge to transfer a circle to any polygon is the Irrational factors that are intrinsically within the shapes (for example, pi, sqrt(2), sqrt(3), …) makes it challenging for a geometer to approximate without using a ruler (scale/measure)…

Any Food for Thought?

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How many-sided polygon would one need to fill in their life?!

Answer: a very large number N-sided polygon, or just a zero – I mean a circle, LIFE itself!!!