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
November 19, 2012 at 10:55 pmIn 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
December 2, 2012 at 12:47 amHow to show that a buckyball has coordinates which can be expressed terms of golden ratio?
Gary Meisner says
December 2, 2012 at 4:22 pmJust 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
December 4, 2012 at 5:52 pmI 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
July 4, 2013 at 5:00 amCould you give me the url of the site?
ShamanAKA11 says
February 8, 2013 at 9:22 pmand this one is a very accurate dodecahedron
http://farm9.staticflickr.com/8531/8457638482_06a7cd2fb6_k.jpg
ShamanAKA11 says
February 21, 2013 at 11:21 amHow to construct a perfect Dodecahedron,Icosahedron and Rhombic Triacontahedron
Dianne Power says
April 14, 2014 at 5:14 pmI 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 ?
Nena says
August 12, 2016 at 3:37 pmCould you share some information on the machinery you are referring to? Is be very interested to know if such machinery exist and how it could be utilized for electricity
Alexis Gelinas says
August 7, 2014 at 2:33 pmAn 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
May 3, 2015 at 8:48 pmThe 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
December 13, 2015 at 3:33 pm1 – 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
December 17, 2015 at 2:07 pmAlready considered. See https://www.goldennumber.net/phipigreatpyramidegypt/
Jesus says
February 2, 2016 at 9:24 amPhi 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
Alex Rd S says
May 10, 2016 at 12:15 amhey in this VIDEO >>>>> https://www.youtube.com/watch?v=Sl0BHueSjvA <<<<<>>>3:11<<<<< the golden number in "angle form" whe the ball hits the oobleck! Why? does a hit like that make the air take that "figure" of a flower!! ……. im guessing that every impact behaves that way? if so….. im thinking in adding a little "fixative" in our physicsmath proyectile ecuations…………. what do you think????????
jreedxxx7@gmail.com
Lucien says
September 5, 2017 at 7:59 amI’m aware that phi exists in the measurements of a square inside a semicircle, but can it be found in the measurements of a square inside a circle?
japet says
August 27, 2018 at 9:20 amwhat is the general formula in getting the golden ratio of a building?
Gary B Meisner says
September 9, 2018 at 10:08 amTo determine the golden ratios found in a building, or in any other object, you can just take one dimension and either multiply or divide it by 1.618. That dimension will be in golden ratio proportion to the original one.
Trần Quang Hùng says
April 16, 2019 at 12:26 amDear friends
I have published an eprint about constructions of Golden ratio in any triangle.
Please see https://arxiv.org/pdf/1904.02011.pdf
Best regards
Tran Quang Hung.
gera says
May 24, 2019 at 9:25 pmbuilding a small circular house with golden ratio (from argentina)
hi dear people, im starting a project of building my circular little house here in argentina and would like to learn about the golden ratio for defining the height of the walls and the height and angles of the triangular roof that will sit on those walls for a diameter of a circle of 6.8 meters. im a bit lost, is it posible that the golden ratio to that diameter would be aprox a roof with the highest point in the center of 4.2 meters aprox? any suggestion will be much appreciated. thanks! gera
Gary B Meisner says
May 25, 2019 at 4:08 pmTo keep things in golden ratio proportion, just take any dimension in question and multiply or divide it by 1.618. There are some special triangles with the legs and base in the ratio of 1.618 to 1, so those may be of interest. There is also a golden ratiobased triangle that appears in the dimensions of the Great Pyramid of Giza, so that may be of interest too. See https://www.goldennumber.net/phipigreatpyramidegypt/. My software site at http://www.phimatrix.com may also give you some ideas.