Pi, Phi and Fibonacci

Phi (Φ) and pi (Π) and Fibonacci numbers can be related in several ways:

  The Pi-Phi Product and its derivation through limits

The product of phi and pi,

1.618033988…  X  3.141592654…,  or  5.083203692,

is found in golden geometries:

Golden CircleGolden Ellipse
Circumference =
Area =


Ed Oberg and Jay A. Johnson have developed a unique expression for the pi-phi product (pΦ) as a function of the number 2 and an expression they call “The Biwabik Sum,”a function of  phi, the set of all odd numbers and the set of all Fibonacci numbers, as follows:


Phi = 2² {1+ [ (2/3) / (F1+F2 Phi) + (1/5) / (F3+F4 Phi) – (1/7) / (F5+F6 Phi) ]
–  [ (2/9) / (F7+F8 Phi) + (1/11) / (F9+F10 Phi) – (1/13) / (F11+F12 Phi) ]
+ [ (2/15) / (F13+F14 Phi) + (1/17) / (F15+F16 Phi) – (1/19) / (F17+F18 Phi) ]
– … }
= 5.083203692….


This relationship was derived after Oberg noticed an interesting relationship between pi and phi while contemplating geometric questions related to the location of the King and Queen’s burial chambers in the Great Pyramid, Cheops, of Giza, Egypt, the design of which is based on phi.You can access the complete paper published by Ed Oberg and Jay A. Johnson, The Pi-Phi Product, in Word, or the Pi-Phi Product in Excel to see their formulation illustrated numerically.


Trigonometric functions relating phi (Φ) and pi (Π)

Divide a 360° circle into 5 sections of 72° each and you get the five points of a pentagon, whose dimensions are all based on phi relationships.

Accordingly, it shouldn’t be too surprising that phi, pi and 5 (a Fibonacci number) can be related through trigonometry:

Phi, the golden ratio, expressed in trigonometric terms

Or, a much simpler way involving, contributed by Dale Lohr:

Pi = 5 arccos (.5 Phi)

Note:  The angle of .5 Phi is 36 degrees, of which there are 10 in a circle or 5 of in pi radians.

Note:  Above formulas expressed in radians, not degrees

Alex Williams, MD, points out that you can use the Phi and Fives relationship to express pi as follows:

5arccos((((5^(0.5))*0.5)+0.5)*0.5) = pi

Robert Everest discovered that you can express Phi as a function of Pi and the numbers 1, 2, 3 and 5 of the Fibonacci series:

Phi = 1 – 2 cos ( 3 Pi / 5)

Pi and Phi in the Great Pyramid of Egypt

Another interesting relationship between Pi and Phi is related to the geometry of the Great Pyramid of Giza.  This relationship connects dimensions of the Great Pyramid to both Pi and Phi, but it is not known with certainty whether this was an intentional aspect of its design, whether its design was based on Pi or Phi but not both, or whether it is a simple coincidence. It relates to the fact that 4 divided by square root of phi is almost exactly equal to Pi:

The square root of Phi (1.6180339887…) = 1.2720196495…

4 divided by 1.2720196495… = 3.14460551103…

Pi = 3.14159265359…

The difference of these two numbers is less than a 10th of a percent.

See the Phi, Pi and the Great Pyramid page for more details.

Pi squared (Π²) and 987

Pi squared (Π²) is 9.8696…, which, if you round to 9.87 and ignore the decimals, is 987, the 17th number of the Fibonacci series. (Contributed by William Erman.)

More on the relationship of Phi squared and Pi

If you’re looking for other interesting ways to relate pi and phi, 6/5 * Phi^2 = 3.1416, which approximates pi. (Contributed by Steve Lautizar.)



  1. says

    Pi squared (Π²) and 987

    Pi squared (Π²) is 9.8696…, which, if you round to 9.87 and ignore the decimals, is 987, the 17th number of the Fibonacci series. (Contributed by William Erman.)

    yes as you also can multiply 21 * 13 and add it to 34 * 21 to make 987 which are the number 7th 8th and 9th of the Fibonacci series.

  2. Chris Weismann says

    PI = 1.2 PHI^2=3.1416

    I found this relationship while studying the work of Rene Schwaller de Lubicz. He had discovered in the measurements of PI and PHI, where both numbers governed it’s PI shape. Over the course of 15 years he measured the Luxor Temple in Egypt and showed that Egypt had a sound knowledge of phi and pi long before the Greeks. Ref his colossal work Temple of Man.
    I then found this website as I wanted to delve into the relationship further.

  3. andries says

    i noticed that pi x (square root phi)= 4 ( or very close) and you can make a rectangular triangle with sizes phi, (square root phi) and 1, both of a charming simplicity.

  4. safari peter says

    From the Giza pyramid one derives the simple relation between pi and phi that is 5/6 of pi minus 1 is phi.

    • says

      That does provide a close approximation of phi. There are endless variations on that approach though, for instance 4 / square root of phi is close to Pi. They’re all interesting, and many are very insightful and creative, but the most meaningful relationships are those that are exact or that can be related to a geometric construction. See this site’s page on Phi, Pi and the Great Pyramid of Giza for other interesting relationships.

  5. says

    I noticed whilst writing a program to draw a ‘flower of life’ that the position of the third (and subsequent) circles, after drawing the first two intersecting circles is related by this formula (forgive me, I’m a computer geek, not a math geek) –

    Given circles of radius r at co-ords X1, Y1 and X2, Y2, the X co-ord of the center of the third circle, X3 = X1 + ( r / (Pi * Phi) ) * Pi.

    Have I restated an existing and known about relationship?
    Interesting site by the way. I’ve written a nice Pascals triangle program that shows the relationship between it and prime numbers that you might be interested in. I’ll upload it to my website if you want to have a look.

  6. u mad ? says

    we noticed that the circumference of the golden circle isn’t pi but phi*pi and we also discovered that the golden eclipse area is pi*phi and that it isn’t pi.

  7. Greg says

    When compared with pythagoras, the angles are 89.1, 31 and 59.9 degrees. Triangle e, phi and pi. Slightly off the 90, 60, 30.


    Maybe calculated on a base of 11 or 9 and not a base of ten???

  8. Armando Hernandez says

    Another trigonometric functinon involving phi is:

    2*sin(pi/10) = phi = 0.618033…

    notice that pi/10 rad = 18 degrees.

    The relation above can be obtained by studying que pentagram or the pythagorean star.

  9. j.Iuliano says

    Here is a geometric conversion approximation for pi to phi, exact to ten decimals:

    (((4*pi^2)-(pi^3))/2)^(3^-1) = phi + sqrt(10^-13)

    This formula shows a counterintuitive result due to the idea that the side areas of a pi cubed structure (4*pi^2) is equivalent to the entire cubic pi volume plus two volume cubes of phi !!?? This equation shows the structure of an elliptic curve:

    ((phi^3)/(pi^3))-(2/pi)+(1/2) = .00000008013

    In the elliptic curve equation, (x^3)+(y^3) = 4*x*y, the xy point, parallel to the x axis tangent point of the elliptic cusp curve generated by the latter equation is, x=1.67894731 , y=2.1165347, can be substituted for x:phi and y:pi, respectively.The interesting thing about surface areas being greater than the sum of cubic volumes is this effect happens to black holes, where 1/4 of the surface area of a black hole contains more information than the entire volume of the black hole!?

  10. Flemming Sejer says

    In the section “Trigonometric functions …”, I find the first note incorrect. It should read: “Note: The angle of Pi/5 is 36 degrees, of which there are 10 in a circle or 5 in Pi radians.” and not “Note: The angle of .5 Phi is 36 degrees, …”

  11. Myrtonos says

    The area of the golden ellipse is simply the pi-phi product multiplied by the square of the semi-minor axis. The convergent of this product with 22 continued fraction values is [5; 12, 53, 2, 14, 1, 1, 4, 2, 5, 11, 3, 11, 3, 1, 4, 4, 17, 1, 45, 5, 2].

    • Gary Meisner says

      That is true, as it is derived from the triangle shown in the article with sides of the sqrt of Phi and 1, with a hypotenuse of Phi

      1/sqrt(phi) = 0.786151377757423
      sqrt(phi) = 1.272019649514070

      atan(1/sqrt(phi)) = 0.666239432492515
      atan(sqrt(phi)) = 0.904556894302381
      Total = 1.570796326794900

      Pi = 3.141592653589790
      Pi/2 = 1.570796326794900

      Pi and Phi can be also related through trigonometry through angles involving the pentagon that divides a circle in five sections.

      Beware though of articles that tell you that Pi * Sqrt (Phi) = 4. That is not true.


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