There are a growing number of claims on the Internet that the value of Pi as 3.14159… is wrong, and that its true value is 3.1446…, which is 4 divided by the square root of the golden ratio, i.e., 4/√φ.
As I’ve been researching the golden ratio since the 1990s, I first heard about this years ago but found the proofs difficult to engage with, and the claim unlikely. It was only in 2022 though that I was finally motivated to investigate this claim in depth, after being contacted by one of the advocates for this new value for Pi.
To see which is the true value of pi, I developed a simple test that requires neither advanced mathematics or geometric constructions. I used the online graphing program Desmos, and then created a YouTube video to present the results in an engaging way.
Here’s a link to the solution on Desmos:
https://www.desmos.com/calculator/0quf8l0zqa
Here’s a link to the presentation on YouTube:
The value of Pi has been investigated since at least the time of Archimedes, who in about 250 BC estimated its area by using polygons that approached the circle’s circumference from both the inside and outside, finding its value to be between about 3.140845 and 3.142857. Zu Chongzhi, a 5th century Chinese mathematician and astronomer, calculated the value of Pi with accuracy to 7 decimal places, between about 3.1415926 and 3.1415927, using a similar method. His calculation remained the world’s most accurate for nearly 1,000 years. Isaac Newton calculated pi to 16 digits of accuracy a year after his invention of calculus in 1665. Godfrey Leibniz and many other found ways to approximate pi using infinite series. (See a complete history of the calculation of Pi.)
The work of all these various methods converged on the same value for pi, but this new claim says that there is an error of approximation in all of them, and that the true value of pi is revealed in elegant geometric constructions which show its value to be 4/√φ. Examples of such sites include Measuring Pi Squaring Phi, Proof of Pi, True Value of Pi and a MathForums site topic on this.
One note of interest on the Desmos model in the video: The blue line “complete coverage area” formula used in the Desmos model in the video has adds a full unit square that extends one unit beyond the radius along each axis. In this way, NO part of the circle’s circumference ever touches the border that defines the “complete coverage area.” The circle of any radius thus fits entirely in the complete coverage area. The circle’s area (a) is less than the complete coverage area (b), which is less than the area defined by π=4/√φ (c). If a < b < c, then how can a = c?
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I appreciate and encourage discussion on this topic so that all can learn and know the truth. As the owner and moderator of this site though, it is my responsibility and right to assure that the comments approved for display here add value to the quality of this post and this site in general.
Please keep the discussion here on topic and focus any comments posted here on the logical and mathematical validity of the method I’ve provided in the YouTube video and Desmos graph above.
I will generally limit discussion on this page to that topic, so that this single page does not duplicate the many years and countless pages of discussion on the value of Pi on other sites intended for that purpose.
I’ll be happy to post links here to such discussion sites that are provided to me where a broader discussions can be had. Here is such an example: MathForums site topic on value of Pi.
The transcription of the script for this video is presented below.
Video Script
Hello everyone,
It’s me, Pi, and I need your help. I’m having a bit of an identity crisis. I don’t know if you’ve heard, but there are some serious allegations being made against me. Now don’t think I’m just being irrational. Here’s what they’re saying:
- My true value is not 3.14159…, but instead 3.1446…., which is 4 divided by the square root of the golden ratio
- This new value is revealed in elegant geometric constructions.
- Archimedes, Newton, Leibniz and all the others who used mathematics to reveal my value all got it wrong because of their errors of approximation.
- Anyone who insists that I’m 3.14159… is in ignorance of the truth and standing in the way of progress.
Well, I want everyone to know the truth. Even if you love me as I am, I want YOU to be able to prove my value to yourself, and I want you to be able to show the truth to even your most mathematically-challenged friends.
So, I’ve created an incredibly simple solution that I think will show which of us is the true Pi.
We’re not going to use advanced mathematics. We’re not going to construct elegant geometries. They’re too time-consuming and too difficult to prove.
Here’s my plan: Let’s just draw a circle on a grid and then count the number of full squares that we need to paint over to completely cover the circle.
This might not be so quick and easy either, but you have more computing power at your fingertips than any of the great mathematicians of the past, so let’s use it.
I’ve created a model to do this test, quickly and easily. It’s available to you in a free online graphing program called Desmos, where you can see how it works and verify the results for yourself.
- The first formula creates the quarter circle in red, based on the circle’s formula, x²+y²=r².
- The next two formulas create a blue line that shows the full squares required to completely cover the quarter circle.
- This summation formula counts the number of full squares enclosed by the
- blue line.
- Next there’s a slider to change the radius r.
So what do we find?
- A quarter circle with a radius of 2 can be covered by painting 6 squares.
- A radius of 5 can be covered by 26 squares.
- A radius of 10 can be covered by 90 squares.
- A radius of 20 can be covered by 335 squares.
So where am I going with this? Well, the difference between me and this new Pi in town is less than 0.1%, 1 part in 1,000.
This means we’re going to need a radius greater than 1000 to get meaningful results. So now that you see how the model works, let’s increase the radius to 2000.
- This quarter circle can be completely covered by painting 3,143,587 squares.
- Let’s add two more formulas.
- This quarter circle’s area as πr²/4 with Pi = 3.141592… is slightly less than this, at about 3,141,592.
- Now let’s calculate the area of the quarter circle as πr²/4 again, but this time using the claimed Pi value of 3.144605…, or 4 divided by the square root of the golden ratio, which is the square root of 5 plus 1 divided by 2.
- The quarter circle’s area according to new Pi is slightly more that the complete coverage area, at about 3,144,605.
So what does this tell us?
This simple model shows us that the area of a circle calculated by this new value for Pi is larger than the area that more than completely covers the circle, which of course is larger than the area of the circle itself.
So what do YOU think? Is it mathematically possible for the area calculated by new Pi to be equal to the area of the circle, or is this new value for Pi of 3.1446… just a false imposter?
I’d love to hear your comments, and to have you share this simple solution with others so that everyone knows the true value of Pi. Thanks for participating in this investigation with me. You’re now in my circle of friends, and I’m infinitely grateful for your support. If you’re ever in the area, stop by and say Pi, or however many of my digits you’ve memorized.
And for the math lovers out there, here’s a little bonus exercise for you. A radius of 2,000 can only reveal so much. What do you discover about the complete coverage area and its relationship to Pi when the radius is 20,000, 200,000, 2,000,000 or more?
And here’s a twist on the coverage formula that increases its border area beyond the circle’s circumference even further. With a radius of 20,000, by how much can you increase the border of the complete coverage area and still have it be less than the area value given if Pi were 4/√φ?
Have fun, and keep seeking and sharing the truth.
My question is does the Geometrical Pi ratio differs from the Mathematical one??? Please respond
I don’t see how it could. A circle is very clearly and simply defined, as is its circumference, diameter, radius and area.
What reason could there be for there to be a different answer for the ratio of the circumfernce and area to the diameter? The mathematics is just an expression of the geometry, not something different or inconsistent.
The use of mathematics and geometry to determine the ratio of the circumference/area to the circle doesn’t change anything about the circle, so it can’t change anything about the ratios of its elements.
Hi Gary!
Let me ask you a question: Why do you keep making things complicated that are, in fact, very simple.
Basically, any method you use that yields the π value of 3.1415926 shows that you are indeed working with straight segments of line. With a Polygone.
When you use a direct method, as I did, measuring π in an exact way, it means in only one step and not infinite ones, you obtain 3.1446.
This again doesn’t mean that 3.1415926 is wrong, it is just te closest value you can obtain approaching the circle with straight lines. What the value 3.1446 really means is, that, evidently, we can’t measure space in the same way with a straight line or with a circular one.
And this should be the point of investigation and not to find out which value wrong.
Personally, I thought my method was quite simple and elegant. It completely avoids the “issue,” if there is one, between straight and curved lines by focusing on the area rather than the circumference. Watch the video and you see that with one simple two-part area formula and a summation formula, I showed this:
For any circle with a radius over about 1400 (to provide enough resolution), if you paint over all the full squares required to COMPLETELY COVER the circle, the number of full squares required is LESS THAN the area that πr² using π=4/√φ says is the area of the circle.
In my example on the video with a radius of 2000, if the entire circle can be COMPLETELY covered by painting 3,143,587 full squares on the XY grid, how could the area of the circle possibly be 3,144,605, especially since the area of the circle is less than the number of full squares required to cover it?
I like where gary is going why do I see something different and everyone else doesn’t?
Don’t know whether to laugh or cry.
Discussions about Giza on Academia site often involve “xxx version of π” because it makes their theories work.
Tried to follow some of the “proofs’ you link to, but will need more time to pick apart. Anything based on the Great Pyramid is ignoring the fact that it’s dimensions were rounded to the nearest cubit. It’s not a perfect Kepler Triangle.
This may be relevant: See Section 5 A new method using Thoth’s Constant, here: https://doi.org/10.5281/zenodo.8105721
People dismissing Archimedes’ method as applied in modern maths, fail to understand the concept of limits. Our polygons have infinite numbers of sides, any “curvature” will be indistinguishable from a straight line. and certainly not enough to change the value of π.
My relevant ruler and compass:
https://www.linkedin.com/pulse/polygonal-method-compared-triangular-quadrature-panagiotis-stefanides/?trackingId=ioa5z4BoRz6lLiTGKr1i9Q%3D%3D
— http://www.stefanides.gr/Html/piquad.html
P.C.Stefanides
http://www.stefanides.gr
Gottfried. You don’t say John Kepler do you.
What a strange story. What would Lenny Euler think. What do they think about his identity, it could cause a crisis.
I recently learned that if you set all the digits so far calculated of either pi or e in 9 point type it would reach to any of the inner planets.
We’ve been studying phi since before we could read. Maths predates writing systems by millennia. I’ve been looking at 16th century Hermeticism and the likes of Dee in particular. I haven’t seen this claim made by any one of that era or earlier.
It’s fascinating that this stuff gains traction.
Hi Gary,
You’ve posted a very clear and straightforward argument for the current value of pi. Thanks for taking the effort to do that as it took some work, Makes a lot of sense.
Is there any reason why there’s not been a physical measurement of pi using modern laboratory technology such as lasers, etc….
That would certainly resolve the issue.
The Value of Pi is an interesting topic. I do not usually comment on such matters online but have studied this in my own way and have applied a system where I could prove a more elegant answer. Firstly this comment is too short to explain it all in detail (a book will be released in 2025) which will.
So I will explain briefly, why the context of a circle is important in finding the correct answer to this connundrum. It is very clear that when we measure a circle ie on paper it has different properties to a circle created within nature, or a standing wave, for example. To measure distance or length, or to measure a wave will create different results. Also in a classical sense what we observe and measure are always approximatations as such.
Let’s just say that my personal opinion on this is that they are both approximations and do not truly define the underlying reality of what pi actually is and how pi functions within the time/frequency domain. For what it is worth I will say that Pi is 3.142696805 (pythagorus pi) and this is correct for 9 decimal places (discreet) , and in so the most accurate Pi that can be used.
So where does this new value of PI leave Euler’s Identity?
There is no such thing as “the ratio of the circumference of a circle to its diameter” which is a 2d concept found nowhere else but in our math books! The correct definition is “Area ratio to its volume” which makes the SQUARE AREA of a circle in 2d be
8/√(2(1+√5)) = 3.1446055110296.
Pi area is pure pseudo area because it is the same “square area” on both a circle and a square! Cred to Jain 108 who also calculated “e” based on the volume of the area which is based on this equation:
e = (7^7/6^6)-(6^6/5^5) = 2.7182621608246
Peace!
No such thing? Found nowhere but in our math books??? The most fundamentally and universally accepted definition for the value of Pi is “ratio of the circumference of a circle to its diameter.”
Take a string and wrap it around the edge a coffee cup, plate or tire. The ratio of the length around the object to the length of its diameter will be the same. That’s Pi, right there in your kitchen and garage.
Space.com shows the Earth’s equatorial circumference and diameter to be 40,075 km and 12,756 km. Their ratio is 3.14165…, within the accuracy for Pi you can expect with 5 decimal places of measurement.
It’s easy to prove that the circle’s area is based on Pi as well, so it’s not a “pseudo area”. See:
https://www.geeksforgeeks.org/how-to-prove-that-the-area-of-a-circle-is-pi-r-squared/
A sphere is just a circle rotated on its diameter, so all these relationships hold between 2D and 3D.
As the YouTube video in this article clearly proves, the claim that Pi is 3.1446… is false.
I can’t get your equation for e to work. I get 2.721464602 on my TI-30Xa and my iPhone. Could you share some of the 5 constituent values with us so that I can check my calculators? 7^7 = 823543, 6^6 = 46656, and 5^5 = 3125. 17.6513846 and 14.92992 are the quotients.
Thanks.
try it in 3 dimensions instead of 2
How or why would that make any difference? A sphere is just a circle rotated on its diameter, so all these relationships hold in 2D and 3D.
>>>>……………… What do you discover about the complete coverage area and its relationship to Pi when the radius is 20,000, 200,000, 2,000,000 or more?………… ….<<<
pls re [ relevant – microcosmos and megacosmos]f:
https://youtu.be/ZXKQFXlWvE0?t=2
If you use the Desmos model, it clearly shows that increasing the radius beyond 20,000 just results in proof of the traditional value of Pi, but to more places of accuracy. At a redius of 2,000,000 the complete coverage area is 3.1415946526×10 to the 12th, which varies only slightly from the value of Pi at 3.1415926536×10 to the 12th. This shows that 4/√φ CANNOT be the true value of Pi.
Pi is contained in base 10 décimal !
80/8.1=pi² about ! =9.876543209876543210
sqrt(9.87)
sqrt(9.8765432)=3.1426 which would be not bad but it’s not a proove just a determined number !
Or roughly ‘decimal’ Phi: 16/1.62 , which of course is the square of 4/√1,62
Hi Gary!
See that interesting comment I found under your Video “A simple Solution”
I’m eagerly awaiting your answer.
If you are going to argue that the Diameter is larger than 112mm, don’t waste your time. It is not longer and to prove it you can construct the polygon with “marbles” or circles of 16mm ø. 22 for the perimeter and 6 for the diameter. Connect the dots/centers of the circles and you get the polygon and the diameter in exact values 352 and 112. Trigonometry seems to be flawed and gives D >112.
*Take a polygon of 22 sides to 16mm each, whose diameter will be 112mm and a perimeter of 352mm.
The ratio btw Perimeter and Diameter will be 352/112=3.142857. Ok?*
*But there is more to this ratio: There is a circle circumscribing the Polygon with the same Diameter 112mmm.
Now let’s calculate the perimeter of the Circle itself using the standard value of π 3.1415926:*
*112*3.1415926=351.858 3712.…….(!)*
….*but, it should have been even larger than 352mmm. Isn’t it?*
*The main problem is not that 3.1446 being too large but that 3.1415926 is too small.*
I dont’ see that post on my YouTube video but here is a reply:
You’ve built a 22-gon using 16mm circles, which gives you a perimeter of 352mm and a diameter of 112mm. From that, you calculated a ratio of 352/112 = 3.142857… and suggested that since this is slightly larger than 112 × 3.141592 = 351.858, then π must be too small.
Here’s what’s important to recognize:
A polygon is not a circle. A 22-gon approximates a circle, but by using straight edges, its perimeter will always be slightly larger than the true circumference of the circle it wraps around.
That’s why your result is larger—and that’s not a flaw in π, it’s a confirmation of how approximations work.
In fact, this is exactly what Archimedes discovered 2,000 years ago: inscribed polygons give too little, circumscribed polygons give too much, and π lies between them.
Your calculation isn’t evidence that π is wrong—it’s a great demonstration of why we need π to calculate the true circumference from a perfect circle, not just from an approximation.
The reason your value is slightly off is not because π is wrong, but because a 22-sided polygon is still not a circle, and no number of marbles can change that.
Hope that helps clarify the math—and again, thanks for joining the discussion!
Sorry.That comment I refer to, is wrong. But it also has nothing to do with the fact that it is a polygon.
The real problem is that there is no 22-sided polygon with a perimeter of 352 and a major diameter of 112.. The diameter is slightly larger than 112..
I didn’t pay enough attention.
Thanks for the follow-up, and I appreciate the clarification.
You’re absolutely right to say that a 22-sided polygon with 16mm sides doesn’t yield a perfect diameter of 112mm. That’s exactly the point—approximating a circle using a polygon, even a regular one, can never perfectly replicate the geometry of a true circle.
In fact, this small discrepancy is exactly what led mathematicians like Archimedes to the discovery of π: by comparing the perimeters of inscribed and circumscribed polygons, he could see that the value of π lies somewhere in between—and that it converges toward a unique, constant value.
The difference you’re noticing—that the diameter is slightly more than 112mm—is a feature, not a flaw. It’s the natural consequence of trying to wrap straight-line segments (a polygon) around a curve (a circle). That tiny gap between polygon and circle doesn’t disprove π—it confirms it.
And that’s the bottom line:
🔹 π is not derived from patterns, polygons, or physical constructions.
🔹 It is defined mathematically as the ratio of a circle’s circumference to its diameter.
🔹 It holds up under trillions of digits of calculation, spaceflight, GPS, quantum physics, and calculus.
So while it’s fascinating to explore geometric approximations and relationships with φ or Fibonacci numbers, none of them change the foundational reality that π ≈ 3.1415926535…. It’s not a matter of belief or perspective—it’s a constant baked into the very structure of mathematics and the universe.
I didn’t say 112+ is a flaw., it is just the ø of that polygon.
🔹 π is not derived from patterns, polygons, or physical constructions.
Well. In fact it is.
🔹 It is defined mathematically as the ratio of a circle’s circumference to its diameter.
It is not exactly like that. When the approximation is made by polygons you can’t define it that way.
To that, 3.1415 is a number that can’t ”rotate”, like a perfect circle. Even by trillions of sides it is still a polygon.
No one is saying that the polygon IS the circle or that its perimeter IS Pi. Talking about the differences between a circle and a polygon is a meaningless defense.
The polygon approach is just one of a NUMBER of MATHEMATICALLY VALID ways to estimate the TRUE value of the circumference of the circle, which when compared to its diameter GIVES us an approximation of Pi.
And you don’t have to go very far in this mathematically valid exercise to see that that ratio is way LESS than 4/√φ.
If you doubt the validity of these approaches please download my Excel model from https://www.goldennumber.net/pi-314159-vs-pi-31446-circumference-solution/. It has a simpler yet estimate of the value of Pi.
Experiment with the input value in cell C5. You only need 500 rows to show that the value of Pi is closer to 3.14159… than 3.14460…
Pump it up to 20,000 rows and truth becomes clearer yet, with a variance from Pi of only 0.00000010394327.
All by using ONLY the formula of a circle and Pythagorean theorem.
Tell me if you can find anything that is incorrect in its assumptions, methods or conclusions.
*No one is saying that the polygon IS the circle or that its perimeter IS Pi. Talking about the differences between a circle and a polygon is a meaningless defense.*
You’re just confirming that 3.1415 is not defined as the ratio btw P and D.
*The polygon approach is just one of a NUMBER of MATHEMATICALLY VALID ways to estimate the TRUE value of the circumference of the circle, which when compared to its diameter GIVES us an approximation of Pi.*
As you say, it is an estimate. The amount of methods yielding 3.1415 as a result boil down basically to only one method of adding an ever growing number of ever smaller straight segments of line. The polygon method.
*And you don’t have to go very far in this mathematically valid exercise to see that that ratio is way LESS than 4/√φ.*
As Plato said back then, “the circle we see is not the real circle”.
The derivation of π 3.1446 as 4b=π is flawless and points out to still unknown characteristics of the Circle and of Space.
Why do you try to set aback these possibilities, is incomprehensible to me.
But the derivation of π 3.1446 as 4b=π is NOT flawless. You make an invalid, unproven assumption with no mathematical validation in your construction that invalidates the results.
And the fact that entire industries rely on the traditional value of Pi to do what they do without catastrophic results should be enough to say to you that maybe, just maybe, there’s an flaw in your construction that you need to see and correct.
The difference between 3.14159… and 3.14460… is huge in this of these applications—managing satellites in orbit, calculating when an eclipse will happen, determining your position with GPS.
As a NASA engineered shared, “If Pi were off we would know it.”
Isn’t it much more incomprehensible yet that thousands of years of mathematical research and countless applications in real world industries would be in error, and your golden ratio based Kepler triangle in a half circle redefines all human research, knowledge and experience on this topic?
Kepler -triangle times π :
(√1×π)² = 9.87 (approx) 16th Fibonacci nr..
(√1,618033…×π)² = 15,97 (approx) 17th Fib.nr.
(√2,618033…×π)² = 25,84 (approx) 18th Fib.nr.
Does not work as neat with that fake Pi. The connection between real Pi and Phi² is actually stronger and more direct (355+22)/(113+7) = 5/6 ×
π (approx). 5/6 × 377/120 = 377/144 (among others)
Thanks for sharing this. I appreciate the curiosity behind finding connections between π, φ, and Fibonacci numbers—it’s a fascinating area. But there are a few key issues with the logic here.
When you write expressions like:
(φ×π)²≈15.97≈17th Fibonaccinumber
You’re essentially multiplying π² (≈ 9.8696) by powers of φ (like φ ≈ 1.618…), then noting the result is close to a Fibonacci number. That may seem “neat,” but it’s numerology, not mathematics. If you try hard enough, you can make almost any constant squared or multiplied by irrational values look like it’s near a Fibonacci number.
Here’s the key: π is not defined by proximity to patterns. It’s defined as the ratio of a circle’s circumference to its diameter—a value that emerges naturally from the very geometry of space. Any connections that rely on manipulating π and φ to land near Fibonacci numbers don’t prove anything—they just fit patterns to predetermined outcomes.
If we reverse-engineer formulas using π, then say “only π fits them well,” that’s circular reasoning, not discovery. The same logic would “prove” dozens of other spurious mathematical constants.
So while the numerical coincidences you’re pointing out are interesting curiosities, they don’t disprove or validate π. The true test of π lies in its consistency across trillions of digits, its use in orbital mechanics, engineering, and physics, and its rigorous derivation from calculus and geometry.
I agree, it is indeed more like numerology, My aim was to trace back where that suspicious relationship between pi and phi is based upon. and to point out that it makes ‘a lot more’ sense to use actual pi instead of adjusting it in order fior the product of root phi and the adjusted pi being 4 (a square)
According to many numerologist it can be found in the great pyramid because of the unit base to twice the height ratio being about root Phi. (14/11)
A lot of esoteric wanna be mathematician state it to represent the squaring of a circle with the peremiter of the unit base of the pyramid being 4.
Nonsense of course. Why not adjust phi instead to make it 4 ?
If one wants to relate 2 definite constants, one should use both.
Their product is just not four, but fourish:
15.97 / 9.87 = 4² / π² (approx)
9.87 / 1.44 = π² / 1.2² (approx)
I (also) just find it fun to figure out such obscure connections. I really like this one (square loop) I sort of created (or knocked off) from the (Kepler) root phi-sort of pi association.
(√1,618−1,272)÷2 = 0,0000031446463138
4/0,0000031446463138=
1272003,144646
4/1272003,144646=0,0000031446463138
Etc..
Makes T. Howard’s ‘impossible’ root2 loop look paile in comparison. (Maybe that’s just an exoression that is used solely in my country)
So my aim was not to define pi alternatively on the basis of (root) phi for some mythical (numerical) reason; in the contrary. Knowing that the square of pi is round about 9.87, which can be used to represent the 17th Fibonacci nr. , it was just meant to be a way of showing how the two constants are numbertheoretically or even mathematically related despite its sort of randomness. The mystical (Egyptian) 4/rootPhi is simply a consequence of taking the square root (out) of the approximation 15,97/9.87. ≈ 4/3.14. or the other way around.
Dividing a square amount of 4 by pi will result in a diameter of a circle of round about root phi. That’s all.
One could alternatively divide 5² by ‘π’ to get a diameter of ≈ 8