Calculating the length of ⅛ of a circle (in order to calculate the total length of the circle).
To prove phew = 3.17157, and cancel phi = 3.14159.

We try to calculate the magnitude of length (1D) with the HELP of the magnitude of area (2D).

For the foundation, let’s take an illustration of a 2×4 plane (length unit × length unit).
If the area is calculated, it will be 2×4 = 8 (area unit)

The lengths of 2 & 4 are valid because of their perfect shape.

The details:
Each point on the line along 2 (length units) if drawn upwards, ALL of them will reach a length of 4 perfectly, so that 2 meets the requirements to be multiplied by 4.

And 4 is also VALID as a full 4, because at all points along the line 4 (length units) a horizontal line can be drawn along 2 (length units).

Likewise, how to calculate the length of the curve along 0 to sin45.

First Stage:
on the curve along 0 to sin45, which is 0.7071068.
At all points on the curve, a vertical line is drawn along 1 (unit of length).

Then there will be an excess in “area 1” outside the 1×1 plane.

then area 1 is MOVED to the empty space under the curve, which is “area 2”.

Then the total area (first stage) obtained is 0.7071068 (unit of area).

Second Stage:

From the tangent point of view, to reach 45°, line X must reach a full length of 1 (unit of length).

Then the remaining line along 0.29289 (unit of length) will be calculated upwards, which is also 0.29289

Why 0.29289 upwards?

Because the cursor along 45° goes up to the position 0.29289 AKA “1-cos45”.

Because the shape is square (equilateral), then the multiplication is valid to be carried out, which is 0.29289 × 0.29289 = 0.08578.

The total area obtained is 0.7071068 + 0.08578 = 0.7928932 (area units).

This area justifies that the length & width have met the requirements to be multiplied.

And the magnitude of the area is translated into the dimension of length, which is 0.79289 (length units).

Total length = 0.79289 × 360/45 = 6.34314 (length units).

And circumference/diameter AKA Phew = 6.34314/2 = 3.17157.