## 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**

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**

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