  NL Squaring the Circle Over the years I have written many articles in which ‘Squaring the Circle’ (StC) played a more or less important role. It was almost always revolving around StC based on circumferences (where the perimeter of the square and the circumference of the circle are identical). I have received countless emails, in which it was pointed out to me that I was wrong. According to the senders, StC is based on surface areas (a square and circle with identical surface area) and not on circumferences. With this article I will show you that there is absolutely no difference between StC based on circumferences and StC based on surface areas.

StC is a magical and mystical phenomenon that has mesmerized many great minds since ancient times. The question is whether it is possible, using only ruler and compass, to construct a circle with exactly the same surface area as a given square. The ruler should only be used to draw lines and not to measure. Over the centuries countless attempts have been made to solve this ‘problem’.
It took until 1882 when Ferdinand von Lindemann proved that the problem is insoluble. Major culprit here is the number Pi. The fact that Pi is a transcendental number, makes it impossible starting with a circle, to construct a square with a surface area equal to that of the circle. Or vice versa, starting with a circle.  StC based on surface areas StC based on circumferences

Countless books have been written on StC. The phenomenon is the central theme of many articles. This almost always involves StC based on surface areas, which has led to the misconception that the insolubility of StC only applies to StC based on surface areas. However, it is also impossible to construct a circle with exactly the same circumference as the perimeter of a given square, with the aid of only one ruler and compass. And again Pi, being a transcendent number, is the limiting factor.
There is absolutely no difference between StC based on surface areas and STC based on circumferences. They are twin brothers (or sisters). The following diagrams will show how easy it is to recognize this.  We start with a square with sides of length A. The surface area of the square is then A squared (A^2). See left diagram above. We now ‘Square the Circle’ based on surfaces. In other words, we make a circle (with radius r) with the same surface area as the square. The formula to calculate the surface area of a circle is pi times the radius squared (pi * r^2). This surface area must be equal to A squared (surface area of the square). It follows that the radius squared (r^2) is equal to A squared divided by pi (A^2/pi). The radius (r) of the circle is equal to A divided by the square root of pi. See right diagram above.  Before we go any further, we calculate the circumference of the red circle. The formula for this is 2 * pi * the radius (r). The radius we calculated in the previous diagram (A divided by the square root of pi). This brings the circumference of the red circle to (2 * pi * (A divided by the square root of pi)). Or 2 times A times the square root of pi. See left diagram above.

Lets continue with squaring the circle. We construct a circle (with radius R) which fits exactly in the initial square. The radius of this circle (R) will, by definition, be equal to half of the side of the square. So R = ½ * A. The area of this circle is pi times R squared, or pi times A squared divided by 4 ((pi * A^2)/4). See right diagram above.  We now have a new square whose surface area is equal to the surface area of the green circle (squaring the circle based on surface areas). Thus the square has a surface area of pi times A squared divided by 4 ((pi * A^2)/4). The side of the square is the square root of this. In other words, it is the square root of ((pi * A^2)/4) which is A times the square root of pi divided by 2. See left diagram above.
With this we have achieved a unique situation. The perimeter of the blue square is 4 times its side. This is 4 times A times the square root of pi divided by 2. Or 2 times A times the square root of pi! See right diagram above. The perimeter of the blue square is equal to the circumference of the red circle. Squaring the Circle based on circumferences! It takes only two simple steps to get from Squaring the Circle based on surface areas to Squaring the Circle based on circumferences. As already mentioned, there is no difference between StC based on surface areas and StC based on circumferences. They are twin brothers (or sisters).
Now we have cleared this persisting misunderstanding, we can return to the deeper questions about Squaring the Circle. For example, the question what Pi really is. Not only the mathematical pi, but also the metaphysical pi. 