Why is the circumference rate equal to 4?

In the previous blog, I shared with you how to calculate the circumference rate. Both the approximate value and the accurate value result are in the 3. 14 ...... but you can see the following picture to deduce the circumference rate.

It means to first draw an external square of the circle, and then convert the square into the n-Edge Shape of the circumference according to the figure, that is, the square is infinitely folded until it is equal to the circumference of the circle! Result π d
= 1*4

π = 4! Er, my outlook on life has completely collapsed. Do you think there are some circles? What's the matter? Thinking...

If π = 4
The area of the circle is π.(2/d) ^ 2 = 1;The square area is also 1! Obviously, this is not correct. Let's take a look at what went wrong?

Let's take a look at the figure below:

This is the first Round Cutting Technique by Chinese mathematician LIU Hui. It establishes an inner polygon in the circle until n is deformed. It is obvious that the length of the Line Segment is equal to the arc length when n approaches infinity.

Let's take a look at the decomposition of the external square:

No matter how it is decomposed, it is eventually a structure composed of D1, D2, and Hu. This is the essence of the error, which is essentially different from the cutover.

But it's not over yet. We default it.D1 + D2
> HuThis fact, but we have to prove it. Let's first switch the problem as follows:

Because d1 = d2 = D3 = D4, the problem is converted to proof.D3
+ D3> HuFor example:

The problem is converted to 1/4 circles, with two radius and longer than the arc length.

When I saw this, I thought about calculus. The proof is as follows:

Set EquationsX ^ 2 + y ^ 2 = 1

This can be written as a parameter equation.

X = cos t

Y = sin t

Tε [0, π/2]

So the circumference is

Hu = records √ (x' (t) ^ 2 + (y' (t) ^ 2) dt

= π/2

I'm dizzy. LZ made the mistake of putting the cart before the horse. What people want is π, And I used π to calculate it again, proving that it failed!

How can we prove this problem without using other methods to find π? LZ is too watery. Please advise me

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Author: Nash _ Welcome to repost. sharing with others is the source of progress!

Reprinted Please retain the original address: http://blog.csdn.net/nash_/article/details/8200349

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