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Square the Circle - 396 x 8 = 3168

While working on the age-old “Squaring the Circle” challenge, I discovered a construction
method that produced a very accurate result.

The challenge is, as accurate as one can be, to construct a square with the same area as
that of a given circle, or visa versa, using only a pair of compasses for drawing circles and a
ruler for drawing straight lines.

I experimented with the interesting numbers 3-6-9, and 396 became the diameter for one circle.
Of course, now that the technique is known, any circle diameter will do.
The square area in my experiment is (a-b 2772 mm x 2) ² = 30735936.
My circle radius result was 3127.86697555... and an area of 30735934.51364089...
It has an accuracy of 99.999995... %.

Unfortunately my drawing program is very basic and has limitations such as:
When zooming in things get distorted fast; Circle perimeters come in segments when you
need them to be perfectly round; Degrees show just a few digits past the decimal, etc.
An accurate drawing program or a very fine pencil point should provide much better results.

Diagram 1


1. Place 8 circles along the X-axis and 2.5 on the Y-axis.
2. Add guide-line A.

Diagram 2

8 x 396

3. Connect a line from b (diagram 1) to c (on the Y-axis), touching the circle at e as accurate
as possible. (Important!)
4. To this b-c line, and circle connection point e, add radius d-e.
5. Add line e-f, it makes the number 7 shape.
(I did not use degrees, I only indicated what they are as found.)

Diagram 3


6. Draw outher circle, begin with radius o-f.
7. Guide line B is a paralel line to d-e (from diagram 2).
8. Zoom in to the 3 small circles.

Diagram 4


9. See diagram 5 for information before returning here.
11. Circle radius l-m. This circle connects to and sits above the small circle.
12. Draw guide line C, from o (diagram 3) to q (connect accurately with q).
13. Draw circle radius h-p giving point n and the diameter for the inner circle.
14. Draw circle radius o-n (diagram 3), giving point v and guide line E.

Diagram 5


10. Circle radius j-i on the B guide line.
This small circle fills "the space between...". Imagine sparkplug gap!

Diagram 6

smallest circle

15. Draw guide line E through v (diagram 3) to guide line A, making intersection point z.

Diagram 7


16. Draw line b to z gives the circle radius.
17. Circle area (radius b-z) ² x π should closely match Square area (line a-b x 2) ²

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