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):

This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?

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!

parthenon golden ratio

Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.

The Actual Value

The Golden Ratio is equal to:

1.61803398874989484820... (etc.)

The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number , and I will tell you more about it later.

We saw above that the Golden Ratio has this property:

a b = a + b a

We can split the right-hand fraction like this:

a b = a a + b a

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

φ = 1 + 1 φ

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:

(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:

φ = 1 2 + √5 2

The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately 0.5 + 2.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

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.)

And here is a surprise: when we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio .

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

Let us try a few:

We don't have to start with 2 and 3 , here I randomly chose 192 and 16 (and got the sequence 192, 16,208,224,432,656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ... ):

The Most Irrational

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

So, it neatly slips in between simple fractions.

Note: many other irrational numbers are close to rational numbers, such as Pi = 3.141592654... is pretty close to 22/7 = 3.1428571...)

No, not witchcraft! The pentagram is more famous as a magical or holy symbol. And it has the Golden Ratio in it:

Read more at Pentagram .

Other Names

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

Footnotes for the Keen

* where did √5/2 come from.

With the help of Pythagoras :

c 2 = a 2 + b 2

c 2 = ( 1 2 ) 2 + 1 2

c 2 = 1 4 + 1

c = √( 5 4 )

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:

φ = −b ± √(b 2 − 4ac) 2a

Using a=1 , b=−1 and c=−1 we get:

φ = 1 ± √(1+ 4) 2

And the positive solution simplifies to:

Kepler Triangle

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

It is really cool because:

Golden Ratio

The golden ratio, which is often referred to as the golden mean, divine proportion, or golden section, is a special attribute, denoted by the symbol ϕ, and is approximately equal to 1.618. The study of many special formations can be done using special sequences like the Fibonacci sequence and attributes like the golden ratio.

This ratio is found in various arts, architecture, and designs. Many admirable pieces of architecture like The Great Pyramid of Egypt, Parthenon, have either been partially or completely designed to reflect the golden ratio in their structure. Great artists like Leonardo Da Vinci used the golden ratio in a few of his masterpieces and it was known as the "Divine Proportion" in the 1500s. Let us learn more about the golden ratio in this lesson.

What is the Golden Ratio?

The golden ratio, which is also referred to as the golden mean, divine proportion, or golden section, exists between two quantities if their ratio is equal to the ratio of their sum to the larger quantity between the two. With reference to this definition, if we divide a line into two parts, the parts will be in the golden ratio if:

The ratio of the length of the longer part, say "a" to the length of the shorter part, say "b" is equal to the ratio of their sum " (a + b)" to the longer length.

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

golden ratio definition

It is denoted using the Greek letter ϕ, pronounced as "phi". The approximate value of ϕ is equal to 1.61803398875... It finds application in geometry, art, architecture, and other areas. Thus, the following equation establishes the relationship for the calculation of golden ratio: ϕ = a/b = (a + b)/a = 1.61803398875... where a and b are the dimensions of two quantities and a is the larger among the two.

Golden Ratio Definition

When a line is divided into two parts, the long part that is divided by the short part is equal to the whole length divided by the long part is defined as the golden ratio. Mentioned below are the golden ratio in architecture and art examples.

There are many applications of the golden ratio in the field of architecture. Many architectural wonders like the Great Mosque of Kairouan have been built to reflect the golden ratio in their structure. Artists like Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this as an attribute in their artworks.

golden ratio examples in nature

Golden Ratio Formula

The Golden ratio formula can be used to calculate the value of the golden ratio. The golden ratio equation is derived to find the general formula to calculate golden ratio.

Golden Ratio Equation

From the definition of the golden ratio,

a/b = (a + b)/a = ϕ

From this equation, we get two equations:

a/b = ϕ → (1)

(a + b)/a = ϕ → (2)

From equation (1),

⇒ a = b

Substitute this in equation (2),

(bϕ + b)/bϕ = ϕ

b( ϕ + 1)/bϕ = ϕ

(ϕ + 1)/ϕ = ϕ

1 + 1/ϕ = ϕ

How to Calculate the Golden Ratio?

The value of the golden ratio can be calculated using different methods. Let us start with a basic one.

Hit and trial method

We will guess an arbitrary value of the constant, then follow these steps to calculate a closer value in each iteration.

For example,

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

Since both the terms are not equal, we will repeat this process again using the assumed value equal to term 2 .

The following table gives the data of calculations for all the assumed values until we get the desired equal terms:

The more iterations you follow, the closer the approximate value will be to the accurate one. The other methods provide a more efficient way to calculate the accurate value.

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

ϕ = 1 + 1/ϕ

Multiplying both sides by ϕ,

ϕ 2 = ϕ + 1

On rearranging, we get,

ϕ 2 - ϕ -1 = 0

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:

ϕ = 1/2 + √5/2

Note that we are not considering the negative value, as \(\phi\) is the ratio of lengths and it cannot be negative.

Therefore, ϕ = 1/2 + √5/2

What is Golden Rectangle?

In geometry, a golden rectangle is defined as a rectangle whose side lengths are in the golden ratio. The golden rectangle exhibits a very special form of self-similarity. All rectangles that are created by adding or removing a square are golden rectangles as well.

Constructing a Golden Rectangle

We can construct a golden rectangle using the following steps:

golden ratio calculation

golden ratio calculation

This is a golden rectangle because its dimensions are in the golden ratio. i.e., ϕ = (√5/2 + 1/2)/1 = 1.61803

What is the Fibonacci Sequence?

The Fibonacci sequence is a special series of numbers in which every term (starting from the third term) is the sum of its previous two terms. The following steps can be used to find the Fibonacci sequence:

Fibonacci sequence is thus given as, 0, 1, 1, 2, 3, 5, 8, 13, 21,.. and so on. Fibonacci sequence and golden ratio have a special relationship between them. As we start calculating the ratios of two successive terms in a Fibonacci series, the value of every later ratio gets closer to the accurate value of ϕ.

The following table shows the values of ratios approaching closer approximation to the value of ϕ. The following table shows the values of ratios approaching closer approximation to the value of ϕ.

☛Related Topics 

Given below is the list of topics that are closely connected to the golden ratio. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

Golden Ratio Examples

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

Note that we are not considering the negative value, as ϕ is the ratio of lengths and it cannot be negative.

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

The golden ratio is a mathematical ratio, commonly found in nature, and when used in a design, it fosters natural-looking compositions that are pleasing to the eye. There are many applications of the golden ratio in the field of architecture. For example, the Great Pyramid of Egypt and the Great Mosque of Kairouan are a few of the architectural wonders in which the concept of the golden ratio has been used. Artists like Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this ratio as an attribute in their artworks. It can be used to study the structure of many objects in our daily lives that resemble a certain pattern

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.

\(F_{15}\) = 377 × 1.618034

≈ 609.99 = 610

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

go to slide go to slide go to slide

how do you solve the golden ratio problem

Book a Free Trial Class

Practice Questions on Golden Ratio

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

The golden ratio is a mathematical ratio that exists between two quantities if their ratio is equal to the ratio of their sum to the larger quantity among the two. In other words, when a line is divided into two parts and the longer part 'a' divided by the smaller part 'b', is equal to the sum of (a + b) divided by 'a', this means the line is reflecting the golden ratio, which is equal to 1.618.

What do you Mean by Golden Rectangle?

Why is the golden ratio beautiful.

The golden ratio is a ratio, which, when used in various fields to design objects, makes the objects aesthetically appealing and pleasing to look at. Therefore, the golden ratio is referred to as a beautiful attribute. It can be noticed in various patterns of nature, like the spiral arrangement of flowers and leaves. There are many applications of the golden ratio in the field of architecture. Many architectural wonders have been built to reflect the golden ratio in their structure, like, the Great Pyramid of Egypt and the Great Mosque of Kairouan.

Why is the Golden Ratio Important?

The golden ratio is a mathematical ratio which is commonly found in nature and is used in various fields. It is used in our day-to-day lives, art, and architecture. Objects designed to reflect the golden ratio in their structure and design are more pleasing and give an aesthetic feel to the eyes. It can be noticed in the spiral arrangement of flowers and leaves.

Where is the Golden Ratio Used in Real Life?

There are many uses of the golden ratio in the field of art and architecture. Many architectural wonders have been built to reflect the golden ratio in their structure. Artists like Leo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this as an attribute in their artworks. It can be used to study the structure of many objects in our daily lives that resemble a certain pattern.

Who Discovered the Golden Ratio?

Ancient Greek mathematicians were the first ones to mention the golden ratio in their work. The 5th-century BC mathematician Hippacus and Euclid contributed a lot of their research work on this subject.

What is Golden Ratio Formula?

The golden ratio formula can be used to calculate the value of the golden ratio. The formula to calculate the golden ratio is given as,

where ϕ denotes the golden ratio.

Study.com

We're sorry, this computer has been flagged for suspicious activity.

If you are a member, we ask that you confirm your identity by entering in your email.

You will then be sent a link via email to verify your account.

If you are not a member or are having any other problems, please contact customer support.

Thank you for your cooperation

Logo white

banner

What is the golden ratio:

Method-1: the recursive method, method-2: the quadratic formula, golden ratio definition:, the golden ratio and the fibonacci numbers, golden ratio in geometry, golden ratio in nature, golden ration in architecture and design, golden ratio in history:, practice questions, golden ratio – explanation and examples.

Golden Ratio

Two quantities $a$ and $b$ with $a > b$ are said to be in golden ratio if $\dfrac{ a + b}{a} = \dfrac{a}{b}$

The ratio $\frac{a}{b}$ is also denoted by the Greek letter $\Phi$ and we can show that it is equal to $\frac{1 + \sqrt{5}}{2} \approx 1.618$. Note that the golden ratio is an irrational number, i.e., the numbers of the decimal point continue forever without any repeating pattern, and we use $1.618$ as an approximation only. Some other names for the golden ratio are golden mean, golden section, and divine proportion.

Golden ration can easily be understood using the example of a stick that we break into two unequal parts $a$ and $b$, where $a>b$, as shown in the figure below

golden section1

Now there are many ways in which we can break the stick into two parts; however, if we break it in a particular manner, i.e., the ratio of the long part ($a$) and the short part ($b$) is also equal to the ratio of the total length ($a + b$) and the long part ($a$), then $a$ and $b$ are said to be in the golden ratio. The figure below shows an example of when the two parts of a stick are in the golden ratio and when they are not.

golden ratio segments

Calculating the golden ratio:

We stated above that the golden ratio is exactly equal to $\frac{1 + \sqrt{5}}{2}$. Where does this number come from? We will describe two methods to find the value $\Phi$. First, we start with the definition that $a$ and $b$ are in golden ratio if

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

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

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

We assume any value for the $\Phi$, lets say we assume $\Phi=1.2$. Now, we put this value in the above formula, i.e., $\Phi = 1 + \frac{1}{\Phi}$ and get a new value of $\Phi$ as follows:

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

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

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

If we keep on repeating this process, we get closer and closer to the actual value of $\Phi$. As we show in the table below

Using the fact that $\Phi = 1 + \frac{1}{\Phi}$ and multiplying by $\Phi$ on both sides, we get a quadratic equation.

$\Phi^2 = \Phi + 1$.

This can also be rearranged as

$\Phi^2 – \Phi – 1 = 0$.

By using the quadratic formula for the equation $\alpha x^2 + \beta x + c = 0$, and noting that $x=\Phi$, $\alpha=1$, $\beta=-1$ and $c=-1$, we get

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

The quadratic equation always has two solutions, in this case, one solution, i.e., $\frac{1  + \sqrt{5}}{2}$ is positive and the second solution, i.e., $\frac{1  – \sqrt{5}}{2}$ is negative. Since we assume $\Phi$ to be a ratio of two positive quantities, so the value of $\Phi$ is equal to $\frac{1  + \sqrt{5}}{2}$, which is approximately equal to 1.618.

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:

There are many interesting mathematical and natural phenomenon where we can observe the golden ratio. We describe some of these below

The Fibonacci numbers are a famous concept in number theory. The first Fibonacci number is 0, and the second is 1. After that, each new Fibonacci number is created by adding the previous two numbers. For example, we can write the third Fibonacci number by adding the first and the second Fibonacci number, i.e., 0 + 1 = 1. Likewise, we can write the fourth Fibonacci number by adding the second and third Fibonacci numbers, i.e., 1+1 = 2, etc. The sequence of Fibonacci numbers is called a Fibonacci sequence and is shown below:

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

If we start dividing subsequent Fibonacci numbers, the results approach closer and closer to the golden ratio as shown in the table below:

Pentagon and pentagram

The golden ratio makes numerous appearances in a regular pentagon and its associated pentagram. We draw a regular pentagon in the figure below.

Pentagon

If we connect the vertices of the pentagon, we get a star-shaped geometrical figure inside, which is called a pentagram, shown below

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,

to name a few.

The golden spiral

Let us take a rectangle with one side equal to 1 and the other side equal to $\Phi$. The ratio of the large side to the small side is equal to $\frac{\Phi}{1}$. We show the rectangle in the figure below.

golden rectangle 1

Now let’s say we divide the rectangle into a square of all sides equal to 1 and a smaller rectangle with one side equal to 1 and the other equal to $\Phi-1$. Now the ratio of the large side to the smaller one is $\frac{1}{\Phi-1}$. The new rectangle is drawn in blue in the figure below

golden rectangle

From the definition of the golden ratio, we note that

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

$\Phi(\Phi -1) = 1$, or

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

Hence, the new rectangle in blue has the same ratio of the large side to the small side as the original one. These rectangles are called golden rectangles. If we keep on repeating this process, we get smaller and smaller golden rectangles, as shown below.

golden spiral 1

If we connect the points that divide the rectangles into squares, we get a spiral called the golden spiral, as shown below.

golden spiral

The Kepler triangle

The famous astronomer Johannes Kepler was fascinated by both the Pythagoras theorem and the golden ratio, so he decided to combine both in the form of Kepler’s triangle. Note that the equation for the golden ratio is

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

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

If we draw a right-angled triangle with hypotenuse equal to $\Phi$, base equal to $\sqrt{\Phi}$ and perpendicular equal to 1, it will be a right-angled triangle. Such a triangle is called the Kelper triangle, and we show it below:

kepler triangle

There are many natural phenomena where the golden ratio appears rather unexpectedly. Most readily observable is the spiraling structure and Fibonacci sequence found in various trees and flowers. For instance, in many cases, the leaves on the stem of a plant grow in a spiraling, helical pattern, and if we count the number of turns and number of leaves, we usually get a Fibonacci number. We can see this pattern in Elm, Cherry almond, etc. However, we must remember that many plants and flowers do not follow this pattern. Hence, any claim that the golden ratio is some fundamental building block of nature is not exactly valid.

It is also claimed that the ideal or perfect human face follows the golden ratio. But, again, this is highly subjective, and there is no uniform consensus on what constitutes an ideal human face. Also, all types of ratios can be found in any given human face.

In the human body, the ratio of the height of the naval to the total height is also close to the golden ratio. However, again we must remember that many ratios between 1 and 2 can be found in the human body, and if we enumerate them all, some are bound to be close to the golden ratio while others would be quite off.

Finally, the spiraling structure of the arms of the galaxy and the nautilus shell is also quoted as examples of the golden ratio in nature. These structures are indeed similar to the golden spiral mentioned above; however, they do not strictly follow the mathematics of the golden spiral.

How much of the golden ratio is actually present in nature and how much we force in on nature is subjective and controversial. We leave this matter to the personal preference of the reader.

Many people believe that the golden ratio is aesthetically pleasing, and artistic designs should follow the golden ratio. It is also argued that the golden ratio has appeared many times over the centuries in the design of famous buildings and art masterpieces.

For example , We can find the golden ratio many times in the famous Parthenon columns. Similarly, it is argued that the pyramids of Giza also contain the golden ratio as the basis of their design.

Some other examples are the Taj Mahal and Notre Damn etc. However, it should be remembered that We cannot achieve the perfect golden ratio as it is an irrational number. Since we are good at finding patterns, it may be the case that we are forcing the golden ratio on these architectures, and the original designers did not intend it.

However, some modern architectures, such as the United Nations secretariat buildings, have actually been designed using a system based on golden ratios.

Similarly, it is thought that Leonardo Di Vinci relied heavily on the use of the golden ratio in his works such as Mona Lisa and the Vitruvian Man. Whether the golden ratio is indeed aesthetic and it should be included in the design of architecture and art is a subjective matter and we leave this matter to the artistic sense of the reader.

If you are indeed interested in using the golden ratio in your works, some simple tips would be to use fonts, such as the heading font and the body text, such that they follow a golden ratio. Or divide your canvas or screen for any painting/pictures/documents so that the golden ratio is maintained.

Once you have used the golden ratio in your work, you will be in a better position to decide the aesthetic value of the golden ratio.

We have discussed the relation of the Fibonacci sequence and the golden ratio earlier. We can find the Fibonacci sequence in the works of Indian mathematicians as old as the second or third century BC. It was later taken up by Arab mathematicians such as Abu Kamil. From the Arabs, it was transmitted to Leonardo Fibonacci, whose famous book Liber abaci introduced it to the western world.

We have already mentioned some ancient structures such as the pyramids of Giza and the Parthenon that are believed to have applied the golden ratio in their designs. We also find mentions of the golden ratio in the works of Plato. Elements is an ancient and famous book on geometry by the Greek mathematician Euclid. We find some of the first mentions of the golden ratio as “extreme and mean ratio” in Elements.

The golden ratio gained more popularity during the Renaissance. Luca Pacioli, in the year 1509, published a book on the golden ratio called divine proportion. Leonardo Da Vinci did the illustrations of this book. Renaissance artists used the concept of the golden ratio in their works owing to its aesthetic appeal.

The famous astronomer Johannes Kepler also discusses the golden ratio in his writings, and we have also described the Kepler triangle above.

The term “Golden ratio” is believed to be coined by Martin Ohm in 1815 in his book “The Pure Elementary Mathematics.”

The Greek letter Phi (i.e., $\Phi$), which we have also used in this article to denote the golden ratio, was first used in 1914 by the American Mathematician Mark Barr. Note that Greek $\Phi$ is equivalent to the alphabet “F,” the first letter of Fibonacci.

More recently, Le Corbusier, the lead architect of the UN secretariat, created a design system based on the golden ratio of the UN secretariat building. In his bestseller book ” The Da Vinci Code, “the fiction writer Dan Brown popularized the myths and legends around the golden ratio in his bestseller book “The Da Vinci Code.”

golden section practice question

Previous Lesson  |  Main Page | Next Lesson

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Geometry (all content)

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

Incredible Answer

Video transcript

InVisionApp, Inc.

Inside Design

A guide to the Golden Ratio for designers

Emily esposito,   •   oct 19, 2018.

G ood design has been up for debate for as long as we’ve been creating. There are endless forums, social media threads, and in-person conversations about what makes for great design, with everyone contributing their own point of view.

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

Top Stories

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

The Golden Ratio is a mathematical ratio you can find almost anywhere, like nature, architecture, painting, and music. When specifically applied to design specifically, it creates an organic, balanced, and aesthetically pleasing composition.

In this article, we’ll dive into what the Golden Ratio is, how to calculate it, and how to use it in design—including a handy list of tools.

What is the Golden Ratio?

Also known as the Golden Section, Golden Mean, Divine Proportion, or the Greek letter Phi, the Golden Ratio is a special number that approximately equals 1.618. The ratio itself comes from the Fibonacci sequence, a naturally occurring sequence of numbers that can be found everywhere, from the number of leaves on a tree to the shape of a seashell.

The Fibonacci sequence is the sum of the two numbers before it. It goes: 0, 1,1, 2, 3, 5, 8, 13, 21, and so on, to infinity. From this pattern, the Greeks developed the Golden Ratio to better express the difference between any two numbers in the sequence.

How does this relate to design? You can find the Golden Ratio when you divide a line into two parts and the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which both equal 1.618. This formula can help you when creating shapes, logos, layouts, and more.

You can also take this idea and create a golden rectangle. Take a square and multiple one side by 1.618 to get a new shape: a rectangle with harmonious proportions.

If you lay the square over the rectangle, the relationship between the two shapes will give you 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.”

If you keep applying the Golden Ratio formula to the new rectangle on the far right, you will end up with an image made up of increasingly smaller squares.

If you draw a spiral over each square, starting in one corner and ending in the opposite one, you’ll create the first curve of the Fibonacci sequence (also known as the Golden Spiral).

101 quotes about design, collaboration, & creativity

How to use the golden ratio in design.

Now that the math lesson is over, how can you apply this knowledge to the work you do on a daily basis?

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

1. Typography and defining hierarchy

The Golden Ratio can help you figure out what size font you should use for headers and body copy on a website, landing page, blog post, or even print campaign.

Let’s say your body copy is 12px. If you multiply 12 by 1.618, you’ll get 19.416, meaning a header text size of 19px or 20px would follow the Golden Ratio and balance the 12px body font size.

If you want to figure out how big your body text size should be, you could do the opposite. If your header text is 25px, you can divide it by 1.618 to find the body text (15 or 16 px).

2. Cropping and resizing images

When cropping images, it’s easy to identify white space to cut out. But, how do you make sure the image is still balanced after you resize it? You can use the Golden Spiral as a guide for the image’s composition.

For example, if you overlay the Golden Spiral on an image, you can make sure that the focal point is in the middle of the spiral.

Leveraging the Golden Ratio can help you design a visually appealing UI that draws the user’s attention to what matters the most. For example, a page that highlights a wide block of content on the left with a narrower column on the right can follow the Golden Ratio’s proportions and help you decide where to put the most important content.

Free icons for product design: The big list

4. logo development.

If you’re designing a new logo and feeling stuck, turn to the Golden Ratio to help you sketch out the proportions and shapes. Many popular logos follow the Golden Ratio, like Twitter, Apple, and Pepsi.

Photo credit: Mostafa Amin and Brandology Studio

Designer Kazi Mohammed Erfan even challenged himself to create 25 new logos entirely based on the Golden Ratio. The result? Simple, balanced, and beautiful icons.

Photo credit: Kazi Mohammed Erfan

Tools to help you use the golden ratio.

You don’t need to break out the pencil and paper to calculate the Golden Ratio — there are a number of apps that can do it for you.

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

Getting started with the Golden Ratio

Once you know what to look for, you’ll start noticing the Golden Ratio everywhere. (Don’t believe us? Look at your hands. Even your fingers follow the Golden Ratio.) The human eye is used to seeing this magical number and we subconsciously react positively to it.

As designers, we can use this number to our advantage. Even small tweaks to the way you crop an image or develop a layout can dramatically improve how your users interact with your design.

Watch it now.

by Emily Esposito

Emily has written for some of the top tech companies, covering everything from creative copywriting to UX design. When she's not writing, she's traveling the world (next stop: Japan!), brewing kombucha, and biking through the Pacific Northwest.

Collaborate in real time on a digital whiteboard Try Freehand

Get awesome design content in your inbox each week, give it a try—it only takes a click to unsubscribe., thanks for signing up, you should have a thank you gift in your inbox now-and you’ll hear from us again soon, get started designing better. faster. together. and free forever., give it a try. nothing’s holding you back..

Golden Ratio Calculator

What is golden ratio, golden rectangle.

The golden ratio calculator will calculate the shorter side, longer side and combined length of the two sides to compute the golden ratio. Before we can calculate the golden ratio it's important to answer the question "what is the golden ratio?". The following section will hope to provide you with an answer.

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

The golden ratio, also known as the golden section or golden proportion, is obtained when two segment lengths have the same proportion as the proportion of their sum to the larger of the two lengths. The value of the golden ratio, which is the limit of the ratio of consecutive Fibonacci numbers, has value of approximately 1.618 1.618 1.618 .

The formula for the golden ratio is as follows. Let the larger of the two segments be a a a and the smaller be denoted as b b b The golden ratio is then ( a + b ) / a = a / b (a+b)/a = a/b ( a + b ) / a = a / b Any old ratio calculator will do this trick for you, but this golden ratio calculator deal with this issue specifically so you don't have to worry!

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

The segment addition postulate calculator can be used to find one of the segment lengths when 3 points are collinear, and two of the distances are known.

The golden rectangle is a rectangle with a length of a + b a+b a + b and width of a a a . This rectangle is often seen in art, as it has been said it's the most pleasing to the eye of all such rectangles. The golden rectangle calculator is a convenient way to find the golden rectangle instead of working it by hand.

The golden ratio is seen in many forms of architecture and in some patterns of nature, such as in the arrangement of leaves in some plants. The golden proportion is also seen in regular pentagons. You can find more information about this shape in the pentagon calculator .

What is the golden ratio?

The golden ratio is a ratio between two quantities that we can also find when we compute the ratio between the sum of these quantities and the greater of the two . Numerically speaking, the number a and b are in the golden ratio if: a/b = (a+b)/a This ratio has a specific value, denoted by the Greek letter φ : φ = 1.618033988749 The golden ratio is highly regarded as figures built following these proportions look particularly pleasant to the human eye.

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

The sides of a golden rectangle with diagonal d = 1 are a = 0.850651 and b = 0.525731 . To find these results:

Why is the golden ratio important?

The golden ratio has always had particular relevance in science and art thanks to its properties and appearance. Talking about math:

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?

Many sources, both historical and contemporary, claim that the golden ratio is rather ubiquitous in Nature. Some examples are:

However, while we can't deny the presence of geometrical patterns in Nature, we can't confirm the exactness of the proportions of the examples above: some present huge variations, while others only approximate the golden ratio.

What is a golden ratio?

Car vs. Bike

Equation of a circle with diameter endpoints, plant spacing.

IMAGES

  1. Golden Ratio Explained: How to Calculate the Golden Ratio

    how do you solve the golden ratio problem

  2. Solving Ratio Problems #5

    how do you solve the golden ratio problem

  3. What is the Golden Ratio?. You know you’re truly geeking out when…

    how do you solve the golden ratio problem

  4. What is exactly the golden ratio?

    how do you solve the golden ratio problem

  5. Cobb Adult Ed Math: Solutions to May 26 Ratio and Proportion Problems

    how do you solve the golden ratio problem

  6. How to Solve Ratio Problems Easily: Try These Tricks!

    how do you solve the golden ratio problem

VIDEO

  1. Golden Ratio

  2. Golden Ratio || Quadratic equation || Super 30 movie

  3. Model of golden ratio

  4. 😇 What is Golden Ratio? Please Help 🤔🤔

  5. What is and how to use the "GOLDEN RATIO"? Season 26

  6. RATIO PROBLEM-SOLVING

COMMENTS

  1. What Are the Six Steps of Problem Solving?

    The six steps of problem solving involve problem definition, problem analysis, developing possible solutions, selecting a solution, implementing the solution and evaluating the outcome. Problem solving models are used to address issues that...

  2. Who Discovered the Golden Ratio?

    The golden ratio was first recorded and defined in written form around 300 B.C. The golden ratio refers to a specific ratio between two numbers which is the same as the ratio of the sum of those numbers to the larger of the two original qua...

  3. What Are Some Examples of Ratio Word Problems?

    An example of a ratio word problem is: “In a bag of candy, there is a ratio of red to green candies of 3:4. If the bag contains 120 pieces of candy, how many red candies are there?” Another example of a ratio word problem is: “A recipe call...

  4. GOLDEN RATIO PROBLEMS WITH SOLUTION

    Learn how to solve golden ratio word problems by watching this video. - You can find all my videos about Mathematics in The Modern World

  5. GOLDEN RATIO SAMPLE PROBLEM

    GOLDEN RATIO PROBLEMS WITH SOLUTION · Fibonacci Sequence and Golden Ratio || Mathematics in the Modern World · Golden Ratio = Mind Blown! · Finding

  6. Defining and Finding the Value of the Golden Ratio

    This video focuses explores the great number Phi, also known as the Golden Ratio. The definition and exact value of the Golden Ratio is

  7. Golden Ratio

    golden ratio (a+b)/a = a/b = 1.618. Have a try yourself (use the slider):. 100. 72. =.

  8. Golden Ratio- Definition, Formula, Examples

    Golden Ratio · Calculate the multiplicative inverse of the value you guessed, i.e., 1/value. · Term 1 = Multiplicative inverse of 1.5 = 1/1.5 = 0.6666... · Step 2:

  9. What is the Golden Ratio in Math?

    The Golden Ratio can be calculated proportionally, using joined line segments AB and BC that obey the Golden Ratio with AB being the shorter

  10. Golden Ratio

    The golden ratio Φ is the solution to the equation Φ 2 = 1 + Φ . Golden ratio examples: There are many interesting mathematical and natural phenomenon where we

  11. The golden ratio (video)

    We get phi is equal to-- do it in orange-- negative b. Well negative negative 1 is 1 plus or minus the square root of b squared. b squared is

  12. How to Calculate the Golden Ratio

    The golden ratio is a famous mathematical concept that is closely tied to the Fibonacci sequence.

  13. The Golden Ratio

    Also known as the Golden Section, Golden Mean, Divine Proportion, or the Greek letter Phi, the Golden Ratio is a special number that

  14. Golden Ratio Calculator

    What is golden ratio · 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