The golden ratio (symbol is the Greek letter "phi" presented at left) is a unique number roughly equal come 1.618

It shows up many time in geometry, art, architecture and also other areas.

You are watching: How to calculate the golden ratio

The Idea Behind It

We find the gold ratio once we division a line into two parts so that:


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Have a try yourself (use the slider):

Beauty

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This rectangle has been made utilizing the golden Ratio, Looks favor a typical frame because that a painting, doesn"t it?

Some artists and architects think the gold Ratio makes the many pleasing and beautiful shape.


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Many buildings and also artworks have the golden Ratio in them, such as the Parthenon in Greece, however it is not really well-known if it to be designed that way.

The really Value

The gold Ratio is same to:

The digits simply keep on going, through no pattern. In truth the golden Ratio is known to be an Irrational Number, and also I will certainly tell you more about that later.

Formula

We saw above that the golden Ratio has actually this property:

ab = a + ba

We can separation the right-hand portion like this:

ab = aa + ba

ab is the golden Ratio φ, aa=1 and also ba=1φ, which gets us:


φ = 1 + 1φ


So the golden Ratio can be defined in regards to itself!

Let us test the using simply a couple of digits the accuracy:


φ =1 + 11.618
=1 + 0.61805...
=1.61805...

With an ext digits we would certainly be much more accurate.

Calculating It

You can use the formula to shot and calculation φ yourself.

First guess that value, then execute this calculation again and also again:

A) division 1 by your value (=1/value)B) add 1C) currently use the value and also start again in ~ A

With a calculator, just keep pushing "1/x", "+", "1", "=", around and also around.

I started with 2 and also got this:


value1/value1/value + 1
21/2 = 0.5 0.5 + 1 = 1.5
1.51/1.5 = 0.666...0.666... + 1 = 1.666...
1.666...1/1.666... = 0.60.6 + 1 = 1.6
1.61/1.6 = 0.6250.625 + 1 = 1.625
1.6251/1.625 = 0.6153...0.6154... + 1 = 1.6153...
1.6153...

It gets closer and closer to φ the an ext we go.

But there are much better ways to calculate it to hundreds of decimal locations quite quickly.

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Drawing It

Here is one method to draw a rectangle v the golden Ratio:

Draw a square of dimension "1"Place a dot half way follow me one sideDraw a heat from that point to an opposite corner

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Now rotate that heat so the it runs along the square"s sideThen girlfriend can extend the square to it is in a rectangle through the gold Ratio!

(Where go √52 come from? watch footnote*)

A Quick means to Calculate

That rectangle above shows united state a an easy formula for the gold Ratio.

When the quick side is 1, the long side is 12+√52, so:


φ = 12 + √52


The square root of 5 is about 2.236068, for this reason the gold Ratio is about 0.5 + 2.236068/2 = 1.618034. This is one easy means to calculate it when you require it.


Interesting fact: the golden Ratio is also equal to 2 × sin(54°), obtain your calculator and check!


Fibonacci Sequence

There is a one-of-a-kind relationship in between the golden Ratio and also the Fibonacci Sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

(The next number is found by adding up the 2 numbers prior to it.)

And here is a surprise: once we take any type of two successive (one ~ the other) Fibonacci Numbers, their ratio is an extremely close to the golden Ratio.

In fact, the larger the pair the Fibonacci Numbers, the closer the approximation. Allow us try a few:


A
B
B/A

2
3
1.5

3
5
1.666666666...

5
8
1.6

8
13
1.625

...
...
...

144
233
1.618055556...

233
377
1.618025751...

...
...
...

We don"t need to start v 2 and 3, here I randomly decided 192 and also 16 (and obtained the succession 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):


A
B
B / A
192
16
16
208
13

208
224
224
432
1.92857143...

...
...
...

7408
11984
1.61771058...

11984
19392
1.61815754...

...
...
...

The most Irrational ...

I think the gold Ratio is the most irrational number. Right here is why ...

We saw before that the golden Ratio have the right to be characterized in regards to itself, like this:
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(In numbers: 1.61803... = 1 + 1/1.61803...)
That have the right to be increased into this portion that goes on for ever (called a "continued fraction"):
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So, it nicely slips in between an easy fractions.

Note: numerous other irrational numbers space close come rational numbers (such as Pi = 3.141592654... Is quite close come 22/7 = 3.1428571...)

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Pentagram

No, no witchcraft! The pentagram is more famous together a wonder or holy symbol.And it has the golden Ratio in it:

a/b = 1.618...b/c = 1.618...c/d = 1.618...

Read an ext at Pentagram.

See more: How Thick Of Steel To Stop A Bullet ? Will 1/4 Plate Steel Stop Bullet

Other Names

The golden Ratio is additionally sometimes called the golden section, golden mean, golden number, divine proportion, divine section and golden proportion.

Footnotes because that the Keen


* where did √5/2 come from?

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With the aid of Pythagoras:

c2 = a2 + b2

c2 = (12)2 + 12

c2 = 14 + 1

c2 = 54

c = √(54)

c = √52


resolving using the Quadratic Formula

We can uncover the worth of φ this way:


Start with:φ = 1 + 1φ
Multiply both political parties by φ:φ2 = φ + 1
Rearrange to:φ2 − φ − 1 = 0

Which is a Quadratic Equation and also we have the right to use the Quadratic Formula:

φ = −b ± √(b2 − 4ac) 2a

Using a=1, b=−1 and also c=−1 us get:

φ = 1 ± √(1 + 4) 2

And the hopeful solution simplifies to:

φ = 12 + √52

Ta da!


Kepler Triangle


We saw above that:φ2 = φ + 1
And Pythagoras claims a right-angled triangle has:c2 = a2 + b2

That inspired a man referred to as Johannes Kepler to produce this triangle:

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It is really cool because:

it has actually Pythagoras and φ together
Nature and The golden Ratio Fibonacci sequence Pentagram Geometry index