## Squaring 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 a circle.

Of course, now that the technique is known, any circle diameter will do.

The square area in my experiment: (**a-b** 2772 mm x 2) ^{2} = 30735936.

My circle radius result: 3127.86697555...,__ with an area of 30735934.51364089...__

I managed an accuracy of at least 99.999995... %

Place 8 circles along the **X**-axis and 2.5 on the **Y**-axis, and add guide-line **A**.

Connect a line starting from **b** (diagram 1) to **c** (on the **Y**-axis), touching the circle at **"e"**

as accurately as possible. Important! __The accuracy of the final result depends on it__.

Draw the line **e-f** parallel to the **X**-axis, and circle radius **o-f**.

Draw circle **1** (radius **u-o**), then **2**, then add guide line **B**.

Circle radius **q-m** as shown (**q** = **B** guide line and circle). It produces point **h**, and the inner circle.

Circle radius **o-h** and draw guide line **C** through point **i**.

Intersection of guide lines **C** and **A** is **z**.

Draw line **b-z** = the circle radius.

Circle area (radius **b-z**) ^{2} x π should closely match Square area (line **a-b** x 2) ^{2}

Doing this by compass and pencil one can get a quite accurate
result,

by drawing the C guide line just below the top of the upper circle.

There are a few variation possibilities to this construction method.

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Lastly, a less accurate but easier and fun construction method where (A-D)^{2} + (D-B)^{2} = (A-B)^{2} :

1. Draw circles 1, 2, 3

2. Draw triangle and post.

3. Draw guide lines as shown.

Circle radius A-B = √10 000 000 mm. Area = π x 10 000 000 (31415926.535... mm)

A-C = 2800 mm, and a square area of (2800 x 2)^{ 2} = 31360000 mm

Accurate to 99.9%