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 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

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.