How to Make a Square Phi…

I learned something the other day – you can find the square root of a number using a semicircle! ( I had forgotten the trignonometric formula: A²  + B² = 1    is that of a circle with a radius 1.)

I also learned that there is a right-angled triangle with sides of √1,√2,√3 – Cool huh?

Anyway it turns out that in just a few simple steps using a ruler and compass (or a convenient graphics program! 🙂 ), using the latter fact, you can ‘construct’ a line of length Phi – which is considered the Golden Mean or Golden Ratio.

  1. Draw 2 perpendicular axes (straight lines) Call the intersection point ‘O’.
  2. Measure one unit up and one unit right on the two axes. Where these points make 2 sides of a square the furthest point of the square to the point O is a distance of √ 2. (Approx 1.4142)
  3. Take this length and replace two opposing sides of the 1 unit square with sides of this new length. The point furthest from the point O is a distance of √ 3.
  4. A line of this length placed at right angles to one end of a line of length √ 2 produces a distance of √5 between their extreme combined line length ends. (See Diagram)


How to make a square Phi

OK – we are almost there! 😉

Take this new line length and add a line of length 1 to it ( √5 + 1).

Find the middle of this line (bisect it with a compass) and draw a semi-circle with that length radius. ( (√5 +1)/2 ). The radius – shown as a purple line on the diagram – has the length Phi ! (Approx 1.618033etc).

Placing this line on the original axes, touching at point O, gives 2 sides of a ‘Square Phi’ !

Phi has some unusual qualities, besides that of being found throughout our natural world in a myriad of ways:

Phi is 1.618033… (it is infinitely long).

Phi Squared = 2.618033… ( Phi x Phi = Phi +1 )

The inverse of Phi = 0.618033… ( 1/Phi = Phi – 1)

The sequence of numbers starting with 1 and multiplying by phi to get each successive number form points on a spiral. If you place lines of this length with one end at the same point for all but place them exactly 72 degrees to each other you make a 5 point ‘star’ with the remaining line end points forming a Phi spiral. The remarkable thing about this is: If you start at the 11th value (the completion of exactly 2 revolutions from the initial starting line) The length of this line is = 11 times the previous value of this line plus the value of the first value for this line. or to put it in a formula: for any line with more than 2 values, the length of the next line on this ‘spoke’ of the spiral ‘wheel’, L (n):

L (n) = L(n-1) x 11 + L(n-2)

for ALL values on any line that meets the starting condition.

Here’s an image that might make it clearer. 🙂

Fibonacci Phi Spiral Pentagram

As you can see the spiral has a direct relationship to one formed by lines of the lengths of the numbers in the Fibonacci sequence, 0,1,1,2,3,5,8,13, etc. If you multiply the fibonacci number by Phi you get an approximation of the next value in the sequence – the values get more exact the higher the number.

And it all comes about from the combination of √1, √2 and √3!





  1. I had totally forgotten ALL of this … until I built my teepee and had to figure out what size carpet I needed. Teepees are not round. They are egg-shaped. You need to figure out how much you need for the round part, then add in the remainder of it. I had to look up all the formulae.

    I did. AND I wound up with a floor covering that was actually the right shape. Well, more or less. After which, I forgot all the formulae again 😀

    Liked by 1 person

    • Beautiful! ( except the forgetting part! – but hey you’ll find it again if you need to, right? 😉
      I never realised teepee’s were ovally based? Egg shaped and conical? Hmmmmm….

      I LOVE the fact that the formula for any circle is given in terms of adding 2 squares!! 🙂 ( and that you can find a square root from a (semi) circle.


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