Pi=3.14159 has been proven incorrect by serous mathematicians. Some of them even wrote books to propose their own value of C/D other than Pi=3.14159.
But their attempt seem not to be successful in gaining attention. So Pi = 3.15157 is still the theorem regarded as the most correct interpretation of C/D.
To figure out the value of Phew, there is the “easy way” and there is also the “difficult way”.
For the easy way, we can draw a square lined diagram in which there is a circle line. At each corners
there is a length from the corner point to the circle line which is called ‘corner diagonal’.
To find the corner diagonal lengths, firstly draw two diagonal lines at the square 2×2. The length between the corner edge points and the circle points equals (diagonal – diameter): 2 = 0.414213
if we look at the square perimeter length, its length equals 4a = 4×2 = 8.
To figure out the length of circumference, you can substract the length of 8 with 4×0.414213, it will be 8 – (4×0.414213) = 6.34314, and this magnitude equals circumference length.
this reality gives the core ratio 6.34314 : 8 = 0.7928932 : 1. This is the ratio of a perfect circle. I call it “& Ratio” ( spelled: “Dan Ratio)
(“Dan” is a word in Indonesian language meaning “and” in English)
“& Ratio” is the Golden Ratio for a perfect circle
Dan ratio can also be viewed
from a 45° angle
In this area, there is a radius with the length of “1” and the arc length
of 0.7928932. So, for 45° ratio of the arc length and the radius equals 0.7928932 : 1
To figure out the value of Pi / Phew we should firstly know “what” really Pi / Phew is. In this case, there are several contexts of Pi or Phew.
The most known one is Circumference over Diameter.
Unfortunately that concept to some extense seems to be hard to imagine unless it’s explained before about its actual idea.
The thing will be different and clear if C/D is understood as two
Firstly, the “circle area”
Seconly, it can also mean “a half of circumference”
The value of both is the same: 3.17157 or for the better precision:
(for calculator) it is 3.1715729.
it depends on its dimension: whether the object has one dimension or two dimensions, i.g. it is “meter” or “meter square”
3.17157, either for 1D or 2D, is the value of Phew. And then again,
the significant thing is “Dan Ratio”
For 1D its frame is a half of 8 –> 4. thus “Dan Ratio” will be multiplied by 4 (length unit). 1/2 Circumference = 4 × 0.7928932 equals 3.17157 (length unit)
The similar way applies for the area of a circle. The frame of the circle area is the square D × D = 2×2 = 4. The circle area equals
Dan Ratio times the frame. The result will be the same, Circular Area = 0.7928932 × 4 = 3.17157
That is the way to figure out Phew with “easy way”
Still in this easy way, there is a much easier way to figure out Phew:
Push all four corner edges till the square line become resembling the circle line: a perfect circle.
Before that pressing, suppose you “cut” the edge points of those four corner, to let some of the length get substracted after the pressing.
By this way, the square perimeter lines (2×4=8) will be shortened as far as 4 × 0.414213
In short: just “press the four corners with hands” as far as 0.414213 each after the square perimeter line are cut at the four corner edge positions
As easy as it is.
Now let’s move to the “hard way” to figure out Phew
For this ‘hard way’ the Phew theorem uses trigonometry, just a simple trigonometry
It takes place at the quarter of the square, which area=1×1, whose area covers 90° of the circular arc. Let’s take just the 45° arc
or triangle area
At the 45° of area, the sine value equals 0.7071068, and value of (1-cos45) equals 0.2928932
For the value of components Sin45 and (1 – Cos45), you can put a square mark for one value, and let another value be as it is. Vis a versa. For instance:
(Sin45)² + (1-Cos45) or (Sin45) + (1-Cos45)²
Mindblowingly, the result will be the same, i.e. 0.7928932.
This is, again, Dan Ratio. The core of a perfect circle as well as its calculation system.