Squaring the Circle with Phi

Squaring the Circle comes within four decimal places using the Golden Ratio.

Even before the foundations of the Great Pyramids were laid men have tried to “square the circle.” That is, in a finite number of steps, construct a square and a circle that are precisely equal in area using only the most primitive instruments; namely, an unmarked compass & straightedge. Some of the greatest men in all of history have attempted to solve this ancient riddle. They have included mathematicians, architects, politicians, artists, musicians, philosophers, astronomers and theologians.

The task was finally “proven impossible” in 1882 when Lindemann showed that pi was a transcendental number. In other words, it cannot be calculated as the root of a polynomial equation with rational coeffecients. Hence, the decimal values of pi are infinite, and since it is not possible to construct the square root of an infinite number, it is therefore “impossible” to square the circle with exact precision. One can only hope to come close.

Christopher Ricci has recently discovered an elegant method which comes about as close as it gets. The technique is extraordinary in that it employs a royal parade of three successive Phi constructions that ultimately converge on the same ratio attained by the well known equation: Phi Squared/5 = Pi/6. The procedure is outlined below.

Download the Squaring the Circle with the Golden Ratio pdf file or visit The Circle is Squared to explore the steps at your own leisure.

If we consider the Red Square as a unit square (Side = 1; Area = 1), the following calculations will result:

Golden Square: Side = Phi (1.618033988); Area = Phi Squared (2.618033986).

Golden Circle: Radius = (0.91287093); Radius Squared = (.833333334). Area = (2.61799388).

With respect to the area, there is virtually no difference between these two shapes. Measured in inches the difference is literally microscopic. And even if we convert them into square feet, the difference would remain undetectable by the naked eye. The area of the Golden Circle subtracted from the area of the Golden Square would be a miniscule .0057751 square inches. Converted to metric = a little over 144 sq. microns. This would enclose an area of 12.111 X 12.111 microns; which is roughly the size of two red blood cells.

As far as linear measurement is concerned, this construction yields a very tight approximation for pi as well; (3.141640784). [Note: The math for this is located on Figure #13 in the pdf file]. This is 99.85% accurate for true pi.  To illustrate just how significant this is we would need to enlarge the shapes astronomically. Imagine, for example, you have a planet with a diameter of a thousand miles. According to pi it would take a car racing along at 60 mph more than 52 hours & 21 minutes to circumnavigate the globe at its equator. If we were to extrapolate our travel time using Phi instead, the difference between the two times would be less than three seconds! Now that’s impressive no matter how you slice it.

The fact that we can attain such a high degree of precision without the aid of modern tools and in so few steps sets this construction apart from some of even the most ingenious techniques. If you have any comments or would like to discuss this further with Chris, you may contact him at [email protected]

Thanks go to for Chris Ricci for his passion and dedication in developing this innovative response to a classic geometric challenge, finding another way to relate phi to pi and for sharing it first with GoldenNumber.net.


  1. says

  2. Rod says

    A new geometry design, offering conceptual proof that only three points are required to square the circle. When the green scalene triangle is created, the length of the horizontal line equates to the square root of Pi:

    (large circle’s diameter = 2000000 units)

  3. Rod says

    A colorful drawing and concise review of the current research plateau:
    http://www.aitnaru.org/images/Pi_Corral.pdf (see design: Impossible Balance)

    “Quick Guide for square Pi” – Simple display of geometric balance in a squared circle.

    The circle (D = 2,000,000 units) is drawn next to last, confirming the starting object: a green perpendicular line set, representing the square root of Pi (1,772,453.850.. units) and half the square root of Pi (886,226.925..).

    The circle is drawn via a perpendicular line set (magenta, perpendicular line not shown). The mid-point distance of the two magenta lines is known when the first object is drawn. When rotated to the precise point on the circle, these lines have length equal to the side of a square inscribed in the golden circle.

    Of course, “stuff” happens along the way but can be guessed from the starting geometry. The red lines, drawn last, confirm all of the good stuff that happened.

    Why “Impossible”? This is not a solution but contemplation of proof …
    and geometrically supportive of the whimsical perspective:
    “To square the circle, one must circle the square.”

    • Rod says

      Continuing research …

      Re: http://aitnaru.org/threepoints.html (a Square One conundrum)

      Perhaps, the circle cannot be squared (according to the Greek rules) but a squared circle might be proven geometrically:

      In the Square One design on this web page, the green scalene triangle (part of the larger isosceles right triangle) contains a 45-degree angle and, theoretically, a base length equal to the square root of Pi. If the circle is squared, the left diagonal side (one end of this red line is attached to the Pi line) must have length equal to a side of a square inscribed in the primary circle.

      And for the square root of Pi, only one diameter provides this exact side length: a circle having a diameter of 2. That Pi is an irrational number is insignificant since the lines in the scalene triangle are complementary (perfectly balance the irrationality of Pi).

  4. Perk Cartel says

    Outstanding, heartiest congratulations to Chris Ricci. As a retired architect and devout practising geometrician my fascination with patterning and mathematics has led to this site. The paradoxical impossibility of squaring the circle yet doing so with such elegance and practical accuracy is a wonderful delight. Thank you so much for sharing this fine work.

  5. says

    Thank you so much Mr. Perk Cartel for your gracious comments. I was wondering if you have had the opportunity to peruse the attached website for further details… http://www.circleissquared.com

    In it I demonstrate how to “square the circle” PRECISELY with only compass & straightedge on Riemannian manifolds. I think you might find it intriguing. Any feedback you may have to offer would be most appreciated. Again, thank you.

    Kindest regards,

    C. Ricci

  6. Philip Edwards says

    What would happen if someone actually succeeded in squaring the circle to infinite precision? Would he be assassinated? I know you’re going to say it’s impossible. My point is: hypothetically, what if someone actually did it?

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