Phi (Φ) and pi (Π) and Fibonacci numbers can be related in several ways:
The product of phi and pi,
1.618033988… X 3.141592654…, or 5.083203692,
is found in golden geometries:
| Golden Circle | Golden Ellipse |
 |  |
| Circumference = p * Φ | Area = p * Φ |
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:
| p 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:
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.)