The value of pi

Sometimes I come up with an idea that I'd like to confirm something obvious known as common sense from scratch.

Today I wanted to make sure why pi is known as 3.1415926535....
The following step is what I took during thinking how to measure pi.

1. The length of the arc of the circle whose radius is 1 should be 2*pi.
2. So pi means the length of the arc of semicircle whose radius is 1.
3. When you divide the semicircle into n pieces of the arc, pi could be calculated approximately by sum of the lengths of chords of pieces.
4. The chord of each arc pieces divided by n should be described as sine((180/n)degrees)/cos((180/2/n)degrees).
5. So I assume that when n gets close to infinity pi could be calculated as the result of n*sine((180/n)degrees)/cos((180/2/n)degrees) .

I tried to put different numbers into n and confirmed the results on google calc.

* the case n=2 : 2.82842712 (actually 2*square root of 2)
* the case n=3 : 3.0 (imagine 3 regular triangles inscribed)
* the case n=10 : 3.1286893
* the case n=100 : 3.14146346
* the case n=1000 : 3.14159136
* the case n=10000 : 3.14159264

Finally in the case n=100000, I could get the value of pi as 3.14159265.

I am content with the result since my assumption looks correct.
But the problem is that again is this article completely irrelevant to 'my Danish life'...