## Golden Ratio

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

## The Idea Behind It

Have a try yourself (use the slider):

Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

Do you think it is the "most pleasing rectangle"?

Maybe you do or don't, that is up to you!

## The Actual Value

1.61803398874989484820... (etc.)

We saw above that the Golden Ratio has this property:

We can split the right-hand fraction like this:

a b is the Golden Ratio φ , a a =1 and b a = 1 φ , which gets us:

So the Golden Ratio can be defined in terms of itself!

Let us test it using just a few digits of accuracy:

With more digits we would be more accurate.

## Calculating It

You can use that formula to try and calculate φ yourself.

First guess its value, then do this calculation again and again:

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

I started with 2 and got this:

It gets closer and closer to φ the more we go.

But there are better ways to calculate it to thousands of decimal places quite quickly.

Here is one way to draw a rectangle with the Golden Ratio:

- Draw a square of size "1"
- Place a dot half way along one side
- Draw a line from that point to an opposite corner
- Now turn that line so that it runs along the square's side
- Then you can extend the square to be a rectangle with the Golden Ratio!

(Where did √5 2 come from? See footnote*)

## A Quick Way to Calculate

That rectangle above shows us a simple formula for the Golden Ratio.

When the short side is 1 , the long side is 1 2 + √5 2 , so:

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

## Fibonacci Sequence

There is a special relationship between the Golden Ratio and the Fibonacci Sequence :

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

(The next number is found by adding up the two numbers before it.)

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

## The Most Irrational

I believe the Golden Ratio is the most irrational number . Here is why ...

So, it neatly slips in between simple fractions.

## Other Names

## Footnotes for the Keen

## Solving using the Quadratic Formula

We can find the value of φ this way:

Which is a Quadratic Equation and we can use the Quadratic Formula:

Using a=1 , b=−1 and c=−1 we get:

And the positive solution simplifies to:

## Kepler Triangle

That inspired a man called Johannes Kepler to create this triangle:

- it has Pythagoras and φ together
- the ratio of the sides is 1 : √φ : φ , making a Geometric Sequence .

## Golden Ratio

## What is the Golden Ratio?

Refer to the following diagram for a better understanding of the above concept:

## Golden Ratio Definition

## Golden Ratio Formula

## Golden Ratio Equation

From the definition of the golden ratio,

From this equation, we get two equations:

Substitute this in equation (2),

## How to Calculate the Golden Ratio?

## Hit and trial method

- Calculate the multiplicative inverse of the value you guessed, i.e., 1/value. This value will be our first term.
- Calculate another term by adding 1 to the multiplicative inverse of that value.
- Both the terms obtained in the above steps should be equal. If not, we will repeat the process till we get an approximately equal value for both terms.
- For the second iteration, we will use the assumed value equal to the term 2 obtained in step 2, and so on.

Since ϕ = 1 + 1/ϕ, it must be greater than 1. Let us start with value 1.5 as our first guess.

- Term 1 = Multiplicative inverse of 1.5 = 1/1.5 = 0.6666...
- Term 2 = Multiplicative inverse of 1.5 + 1 = 0.6666.. + 1 = 1.6666...

Another method to calculate the value of the golden ratio is by solving the golden ratio equation.

The above equation is a quadratic equation and can be solved using quadratic formula:

ϕ = \(\frac{-b \pm \sqrt{ b^2 - 4ac}}{2a}\)

Substituting the values of a = 1, b = -1 and c = -1, we get,

ϕ = \(\frac{1 \pm \sqrt{( 1 + 4 )}}{2}\)

The solution can be simplified to a positive value giving:

## What is Golden Rectangle?

## Constructing a Golden Rectangle

We can construct a golden rectangle using the following steps:

- Step 2: Using this line as a radius and the point drawn midway as the center, draw an arc running along the square's side. The length of this arc can be calculated using Pythagoras Theorem : √(1/2) 2 + (1) 2 = √5/2 units.
- Step 3: Use the intersection of this arc and the square's side to draw a rectangle as shown in the figure below:

## What is the Fibonacci Sequence?

- We start by taking 0 and 1 as the first two terms.
- The third term 1, is thus calculated by adding 0 and 1.
- Similarly, the next term = 1 + 2 = 3, and so on.

## Golden Ratio Examples

Example 1: Calculate the value of the golden ratio ϕ using quadratic equations.

Example 2: What are the different applications of the golden ratio in our day-to-day lives?

Example 3: The 14 th term in the sequence is 377. Find the next term.

We know that 15 th term = 14 th term × the golden ratio.

Therefore, the 15 th term in the Fibonacci sequence is 610.

go to slide go to slide go to slide

## Practice Questions on Golden Ratio

Faqs on golden ratio, what is the golden ratio in simple words.

## What do you Mean by Golden Rectangle?

Why is the golden ratio beautiful.

## Why is the Golden Ratio Important?

## Where is the Golden Ratio Used in Real Life?

## Who Discovered the Golden Ratio?

## What is Golden Ratio Formula?

where ϕ denotes the golden ratio.

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## What is the golden ratio:

## Calculating the golden ratio:

$\frac{a}{b} = \frac{a + b}{a} = 1 + \frac{b}{a}$

Let $\Phi = \frac{a}{b}$ then $\frac{b}{a} = \frac{1}{\Phi}$, so the above equation becomes

$\Phi = 1 + \frac{1}{1.2} = 1.8333$.

Now, we put this new value again in the formula for the golden ratio to get another value, i.e.,

$\Phi = 1 + \frac{1}{1.8.3333} = 1.54545$.

This can also be rearranged as

$\Phi = \frac{1 \pm \sqrt{1- 4 \times 1 \times -1}}{2} = \frac{1 \pm \sqrt{5}}{2}$.

Using the above discussion, we can define the golden ratio simply as:

The golden ratio $\Phi$ is the solution to the equation $\Phi^2 = 1 + \Phi$.

## Golden ratio examples:

$0, \,\, 1, \,\, 1, \,\, 2,\,\, 3,\,\, 5,\,\, 8,\,\, 13,\,\, 21,\,\, 34, \cdots$

## Pentagon and pentagram

Many lines obey the golden ratio in the above figure. For example,

$\frac{DE}{EF}$ is in golden ratio

$\frac{EF}{FG}$ is in golden ratio

$\frac{EG}{EF}$ is in golden ratio

$\frac{BE}{AE}$ is in golden ratio,

$\frac{CF}{GF}$ is in golden ratio,

## The golden spiral

From the definition of the golden ratio, we note that

$\Phi^2 -\Phi -1 = 0$, we can rewrite it as

$\frac{\Phi}{1} = \frac{1}{\Phi -1}$

## The Kepler triangle

It is similar in format to the Pythagoras formula for the right-angled triangle, i.e.,

$\textrm{Hypotenuse}^2 = \textrm{Base}^2 + \textrm{Perpendicular}^2$,

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## Geometry (all content)

Unit 1: lesson 6, the golden ratio, want to join the conversation.

## Video transcript

## A guide to the Golden Ratio for designers

Emily esposito, • oct 19, 2018.

That’s the beauty of design, right? Everyone can interpret it differently.

## Top Stories

## What is the Golden Ratio?

“While there will never be a one-size-fits-all approach for designing, there is a concrete, mathematical approach that can help us get one step closer to creating amazing design experiences every time: the Golden Ratio.”

## 101 quotes about design, collaboration, & creativity

How to use the golden ratio in design.

Here are four ways to use the Golden Ratio in design:

## 1. Typography and defining hierarchy

## 2. Cropping and resizing images

## Free icons for product design: The big list

## Photo credit: Mostafa Amin and Brandology Studio

## Photo credit: Kazi Mohammed Erfan

Tools to help you use the golden ratio.

Here are five tools to help you use the Golden Ratio in your designs:

- Golden Ratio Calculator: Calculate the shorter side, longer side, and combined length of the two sides to figure out the Golden Ratio.
- goldenRATIO : Created for designers and developers, this app gives you an easy way to design websites, interfaces, layouts, and more according to the Golden Ratio. It includes a built-in calculator with visual feedback and features to store screen position and settings, so you don’t have to rearrange the Golden Ratio for every task.
- Golden Ratio Typography Calculator : Discover the perfect typography for your website by entering your font size and width. You can optimize based on font size, line height, width, and characters per line.
- PhiMatrix : This Golden Ratio design and analysis software comes customizable grids and templates that you can overlay on any image. It can be used for design and composition, product design, logo development, and more.
- Golden Ratio Sketch resource : Download a free Sketch file of the Golden Spiral to help with image and layout composition.

## Getting started with the Golden Ratio

## by Emily Esposito

## Collaborate in real time on a digital whiteboard Try Freehand

## Golden Ratio Calculator

What is golden ratio, golden rectangle.

You can also check out the proportion calculator if you want to analyze ratios in general.

Here's a step by step method to solve the ratio by hand.

- Find the longer segment and label it a a a
- Find the shorter segment and label it b b b
- Input the values into the formula.
- Take the sum a a a and b b b and divide by a a a
- Take a a a divided by b b b
- If the proportion is in the golden ratio, it will equal approximately 1.618 1.618 1.618
- Use the golden ratio calculator to check your result

## What is the golden ratio?

## What is the length of the sides of a golden rectange with diagonal 1?

- Use Pythagoras' theorem to find the length of the side b as a function of a : `b = sqrt(1 - a²).
- Compute the length of the side a knowing that a/b = φ : a/b = φ a/sqrt(1 - a²) - φ a = sqrt(φ²/(1 + φ²)) = 0.850651
- Compute the length of side b with the following formula: b = a/φ = 0.525731

## Why is the golden ratio important?

- A golden rectangle (a rectangle whose sides are in golden ratio) can be split into two smaller golden rectangles (it maintains its proportions).
- The golden ratio deeply correlates with the number 5 . This number appears in its definition ( φ = (1 + √5)/2 ) and the pentagon as the ratio between diagonal and side.

In arts, the golden ratio appeared more recently: Dalí, for example, used this ratio in its works.

## Where can I find the golden ratio in Nature?

- The growth pattern of leaves;
- The geometrical surfaces of some vegetables and shells;
- The proportions of some animals' bones.

## Car vs. Bike

Equation of a circle with diameter endpoints, plant spacing.

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