Phi and Geometry

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)

Phi, the Golden Ratio, construction with three lines

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.

Phi, the Golden Ratio, construction with a triangle in a circle

4 sides: Square

Insert a square inside a semi-circle.  The ratio of AG to AB is Phi.

Phi, Golden Ratio, construction with a square in a circle

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.

Phi, Golden Ratio, construction with a pentagon in a circle

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.

Pyramid based on phi, the golden proportion

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.)

Forming a golden rectangle based on phi, the golden ratio

Construction of 1/Phi (phi) showing golden ratios

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

Phi, the goldenn ratio, expressed in trigonometric terms

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:

Dodecahedron / Icosahedron from 3 golden rectangles based on phi, the golden proportion

Do it yourself dodecahedron from 3 golden rectangles based on phi, the golden ratio

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 based on phi, the golden ratio


icosahedron based on phi, the golden ratio

Face shapePentagonTriangle

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 Quasi-crystals pages.


  1. Dejan says

    In a right triangle
    c^2=a^2 + b^2
    but only when a=2*b then
    and a=(c+b)/phi
    c=√(2^2+1^2)=√(4+1)=√5=2.23606798 or
    c= 2*1.618033989-1=3.23606798-1=2.23606798

  2. 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.

  3. 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 ?

  4. 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

    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 odd-sided regular polygons?

  5. 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.

Leave a Reply

Your email address will not be published. Required fields are marked *