Phi (Φ) was described by Johannes Kepler as one of the “two great treasures of geometry.” (The other is the Theorem of Pythagoras.)
Phi appears in many basic geometric constructions.
3 lines:
Take 3 equal lines. Lay the 2nd line against the midpoint of the 1st. Lay the 3rd line against the midpoint of the 2nd. The ratio of AG to AB is Phi, the Golden Ratio. (Contributed by Jo Niemeyer)
3 sides: Triangle
Insert an equilateral triangle inside a circle, add a line at the midpoint of the two sides and extend that line to the circle. The ratio of AG to AB is Phi.
4 sides: Square
Insert a square inside a semicircle. The ratio of AG to AB is Phi.
5 sides: Pentagon
Insert a pentagon inside a circle. Connect three of the five points to cut one line into three sections. The ratio of AG to AB is Phi.
When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry shown below creating an angle of 51.83 degrees, the cosine of which is phi, or 0.618.
A ruler and compass can be used to construct the “golden rectangle,” as shown by the animations below, which was used by the Greeks in the Parthenon. (See also the Orthogons page.)
Phi also defines other dimensions of a pentagon.
There are also a number of geometric constructions using a circle which produce phi relationships, as shown on the Geometric Construction of Phi in Circles page.
Phi can be related to Pi through trigonometric functions
Note: Above formulas expressed in radians, not degrees
Phi appears in 3D geometric solids
Take three golden rectangles and assemble them at 90 degree angles to get a 3D shape with 12 corners:  Click on the shape below and the print the page to do it yourself:  
 
This is the basis for two geometric solids  
The 12 corners become the 12 centers of each of the 12 pentagons that form the faces of a dodecahedron.  The 12 corners can also become the 12 points of each of the 20 triangles that form the faces of a icosahedron.  
Dodecahedron
 Icosahedron
 

Some interesting aspects of dodecahedrons and icosahedrons:
A dodecahedron with sides of length 1 embeds a cube with sides of length is Phi.
An icosahedron with sides of length 1, the dual dodecahedron has sides with length 1/Phi. In other words, the dual of the dodecahedron with side of length 1 is an icosahedron with sides of length Phi.
Learn more about phi and geometry on the Penrose Tiling and Quasicrystals pages.
Dejan says
In a right triangle
c^2=a^2 + b^2
but only when a=2*b then
c=a*phib
and a=(c+b)/phi
examle
a=2
b=1
c=√(2^2+1^2)=√(4+1)=√5=2.23606798 or
c= 2*1.6180339891=3.236067981=2.23606798
(c+b)/phi=a
(2.23606798+1)/1.618033989=3.23606798/1.618033989=2
Roopak Tamboli says
How to show that a buckyball has coordinates which can be expressed terms of golden ratio?
Gary Meisner says
Just see the Bucky Ball page for details and a complete list of the xyz coordinates in 3D space of all its corners and points.
Charles Groff says
I found the proof for the “3 Lines” example provided by Jo Niemeyer on another site. Jo is indeed correct, and yet again, PHI appears out of nowhere. Amazing.
Gayatri says
Could you give me the url of the site?
ShamanAKA11 says
and this one is a very accurate dodecahedron
http://farm9.staticflickr.com/8531/8457638482_06a7cd2fb6_k.jpg
ShamanAKA11 says
How to construct a perfect Dodecahedron,Icosahedron and Rhombic Triacontahedron
Dianne Power says
I have nil to minimum knowledge about this but Had a dream many years ago and obsessed about what the shape was which I saw. I now know that this was it!! Thank you for that . could it be related to a piece of machinery which may be used in producing propulsion or electricity ?
Alexis Gelinas says
An interesting oberservation I made last week:
In a regular pentagon with side length=1, if you connect all the diagonals, you create a smaller pentagon within the original one, and the side length of this smaller pentagon = Phi / (1 + Phi).
This relationship also holds with regular heptagons, except in that case, there are two different diagonal lengths (I call the smaller diagonal “rho” and the larger diagonal “sigma” after this paper http://sylvester.math.nthu.edu.tw/d2/imotraininggeometry/regpoly.pdf)
In this case, the regular heptagon has side length=1, and the smaller heptagon has side length = Sigma / (1 + Sigma).
Could this be generalized for all oddsided regular polygons?
john gury says
The inscribed pentagon example is not stated correctly. It needs to be a regular pentagon. period. It is irrelevant that it is inscribed in a circle.. Also, “Insert a pentagon inside a circle” is meaningless as there are infinite pentagons to insert, which should be inscribe.
Here is an interesting extension of the equilateral triangle example that extends the phi cuts with a regular pentagon. https://drive.google.com/open?id=0B86gMl7IvdErVkd4TjJtUE9BY00&authuser=0
Mike Anderton says
1 – phi = 1 / phi
is I guess you’d call one of the “identity formulas” for phi.
If we take the inverse cosine of 1 – phi to get the angle this represents, the answer is
51.827 degrees, which is extremely close to the angle of the sides of the Great Pyramid at Giza, within a few hundredths of a degree.
Something to consider?
Gary B Meisner says
Already considered. See http://www.goldennumber.net/phipigreatpyramidegypt/
Jesus says
Phi angle in the Great Pyramid, is a minimum perimeter triangle:
http://triancal.esy.es/?lang=en&y=2.5980762113533&z=2.5980762113518&p=8.65208917663
Secret rooms are situated in notable points as othocenter:
http://triancal.esy.es/HTML/GG.html?C=0.90455689430153&B=0.90455689430153&a=3.3048039917787&lang=en