As easy as Pi? A simple solution to a challenging problem.
With the growing number of claims on the Internet that Pi is not 3.14159… but rather 3.1446… (4/√φ), I set out to find the simplest possible way to determine which value is correct. As someone known for my work with Phi (φ), this claim, which links Pi to Phi, naturally deserved my attention.
While dozens of mathematicians, dating back to Archimedes, have already provided rigorous proofs for the value of Pi, these methods are often inaccessible to the average person. This makes it challenging for people to independently verify claims, especially when they involve complex mathematical concepts. After all, who among us—whether walking down the street or sitting in a high school math class—can confidently tackle Pi using calculus, infinite series, geometric approximations, Monte Carlo methods or Fourier series? And let’s be honest, who doesn’t love the appeal of a good conspiracy theory?
You’ll find countless claims on the Internet that attempt to “prove” Pi equals 3.14460… using intricate geometric constructions. However, for the average person, validating these claims is no easier than tackling the traditional mathematical methods. Carefully working through all the steps of these constructions, and identifying the flawed assumptions or logical errors, can take hours—if not days.
Pi = 3.14159… vs Pi = 3.1446… – A simple solution
In my other article on this topic, I used an incredibly simple approach to test the value of Pi: I drew a circle on a grid and counted how many squares it took to completely cover the circle.
This simple method only became possible with the availability of personal computing power, so it was never used by mathematicians of earlier eras.
In this model, the area based on the true value of Pi must be less than the total number of squares needed to completely cover the circle. So, an area based on any claimed value of Pi that exceeds the number of squares required to completely cover the circle must be false.
Since the difference between 3.14159… and 3.14460… is less than 1 part in 1,000, you only need a circle with a radius slightly larger than 1,000 to prove which value is correct.
You can explore the details in my article, Pi = 3.14159… vs Pi = 3.1446… – A Simple Solution, where I also provide a Desmos model and a YouTube video demonstrating how it works.
This method clearly proved that the traditional value of Pi is correct. However, even after this conclusive result, some supporters of the Pi = 4/√φ theory didn’t give up. They then claimed that there are actually two values for Pi: one for the area of a circle and another for its circumference. Let’s take a closer look at why that idea doesn’t hold up either.
Proving Pi’s relationship to Circumference and Area
Let’s tackle this from two angles: first by understanding it conceptually, and then by proving it through empirical methods.
Here’s what we already know about circles:
- The radius of a circle (r) is half its diameter (D), so 2r = D.
- The area of a circle (A) is Pi times the square of the radius, or A = πr².
- The circumference of a circle (C) is Pi times its diameter, or C = πD, which can also be written as C = 2πr.
But why exactly does the area of a circle equal πr²? Why must this be the same Pi?
The answer is actually quite simple. The area of any shape is calculated by multiplying its height by its width. If you imagine cutting a circle into very fine slices, you can rearrange these slices to form a shape that looks like a rectangle. From there, the area becomes as straightforward as multiplying height times width, just as with any other geometric shape.
This concept is beautifully demonstrated in this YouTube video by MathematicsOnline:
Area is simply Height times Width!
It’s easy to see how the rectangle we use to calculate the area of the circle is created. The height of the rectangle is simply the circle’s radius (r). The length of the rectangle comes from half of the circle’s circumference, since we divide the circle into slices with half on the top and half on the bottom. Since the circumference is 2πr, half of that is πr.
So, the area of the rectangle becomes height times width, or r times πr, which equals πr².
Conceptually, this shows that the same radius (r) is used to calculate both the area and the circumference of a circle. Therefore, the value of Pi that determines the circumference is the same value of Pi used to calculate the area.
Since Pi is used in both circumference and area calculations, it’s essential that the same value is applied consistently to both, as they are interconnected in a circle’s geometry.
That should be enough to demonstrate that there can only be one value for Pi, but let’s take it a step further and prove it empirically.
Emperical proof for π=3.14159… for the circle’s circumference
I’m still working on a way to demonstrate this using a Desmos graphing calculator model, but for now, let’s explore how we can get to the same solution and proof using an Excel model.
The model is quite simple. If you’re familiar with the Pythagorean theorem (a² + b² = c²) for right triangles, you’ll easily understand it. You can visually check the accuracy of each cell in the Excel spreadsheet by looking at the formulas or using a basic calculator.
This model uses the Pythagorean theorem to calculate the x and y coordinates for every point along the circle’s circumference, based on the equation of a circle: x² + y² = r².
It then calculates the length of the hypotenuse (the line that follows the curve of the circle) for each point on the circumference, where the y-value is an integer.
Excel’s 15 places of accuracy is more than enough
I can understand that some might say, “But that’s just an estimate, not the exact value of Pi.” And technically, you’re right—it’s an estimate. But let me reassure you that this method is far more accurate than what’s necessary to address the Pi debate.
The traditional Pi value of 3.14159… and the alternative claim of 3.14460… only start to differ noticeably at the 3rd decimal place. However, with Excel, we calculate the hypotenuse lengths to 15 decimal places, which provides an extremely precise estimate. This gives us a level of precision far beyond what’s needed to determine an accurate estimate for the circumference.
Additionally, we can make these estimates as precise as we want by using larger radii. The larger the radius, the smaller the difference between the arc of the circle and the hypotenuse used to measure it. Below are the results of the model using radii of 200, 2000, and 20,000:
As you can see, even with the “low resolution” version using a radius of 200, the Pi estimate of 3.14148… differs from the traditional value by only 0.0001. When we increase the radius to 20,000 for higher accuracy, the Pi estimate of 3.14159… varies by only 0.0000001.
The Excel model used a radius of only 20,000 to keep the file size small. If you copy and paste the last row to extend it to a radius of 1,000,000, you’ll find that it produces as estimate for Pi of 3.14159265329582, which varies from the traditional value of Pi by only 0.00000000029398!
This level of precision offers conclusive empirical evidence that the value of Pi as 3.14460… simply cannot be correct.
The Excel model used can be downloaded here:
Pi_circumference_estimate_Excel_model_Gary_Meisner_2024.xlsx
You can also view a screen shot of the model and the formulas used here:
Moving beyond the Pi = 4/√φ myth
If you’ve been wondering whether the traditional value of Pi, 3.14159…, could be incorrect, I hope you find the simple proofs presented here helpful, easy and conclusive. You can validate it yourself: for the circumference, use the model described above, and for the area, check out the Desmos model in my article at Pi = 3.14159 vs Pi = 3.1446 – A Simple Solution. It’s important to note that neither model uses Pi in the calculations—both are based purely on applying the Pythagorean theorem to the circle’s formula.
In every proof I’ve examined for π = 4/√φ, there have been flaws—usually due to incorrect assumptions or logical errors. If you’ve been involved in supporting or developing proofs for Pi = 4/√φ, please know that revisiting these ideas is part of the scientific process. There’s no harm in learning and improving—math is about exploring and refining our understanding.
The most common mistake I’ve seen is an unsupported assumption that equates properties of squares and circles, which simply isn’t valid. The difference between 3.14159… and 3.14460… is less than 0.1%, and it’s not something you can detect with the naked eye. This makes it easy to be misled by geometric constructions, which is why verifying with proper mathematics is so important.
I understand that many people working on these proofs might not have a deep mathematical background, and that’s okay—it’s a learning process. I encourage you to review your work carefully, validate your results with well-established methods, and continue expanding your understanding of math. It’s crucial to approach this with an open mind and a commitment to accuracy.
If you still believe that π=4/φ, it’s critical to realize this:
You can’t just do a geometric construction or even a physical measurement that you claim proves your case. You have to explain why every other method ever used to calculate the value for Pi failed. Calculus, infinite series, Monte Carlo simulations and other mathematical methods are used in countless industries for countless applications and all produce accurate results wherever they area applied. How can these methods get it right for every other problem to which they are applied, yet fail for something as simple as the value of Pi? How can that make any sense? Move beyond the concerns about the accuracy of Archimedes and his polygons and take on Newton and calculus and the fact that we can now create very simple computer programs and spreadsheet models that show the value of Pi to 15 digits within a few minutes from our home computers. How can you show the failings in all of these other methods?
As for those influenced by claims that physical measurements show Pi to be something other than 3.14159…, it’s important to recognize that such small differences require extremely precise equipment. Basic measurement tools used in these “proofs” aren’t sufficient to challenge centuries of work, especially when multiple mathematical approaches all confirm the same value for Pi.
It’s time to move beyond the idea that π = 4/√φ and direct our energy toward projects that deepen our understanding and lead to meaningful discoveries.