golden ratio problem solving examples

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:

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 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),

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.

• 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.

For example,

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...

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:

• Step 1: First, we will draw a square of 1 unit. On one of its sides, draw a point midway. Now, we will draw a line from this point to a corner of the other side.

• 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:

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:

• 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.

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.

• Fibonacci Numbers
• Quadratic Equations
• Ratio, Proportion, Percentages Formulas

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

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.

• Mathematicians
• Math Lessons
• Square Roots
• Math Calculators
• Golden Ratio – Explanation and Examples

JUMP TO TOPIC

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.

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

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.

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.

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

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.

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

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.

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

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:

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.”

Previous Lesson  |  Main Page | Next Lesson

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!

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 then do substitutions like this:

a b = a a + b a ↓      ↓      ↓ 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.

Powers (Exponents)

Let's try multiplying by φ :

φ = 1 + 1 φ ↓     ↓     ↓ φ 2 = φ + 1

That ended up nice and simple. Let's multiply again!

φ 2 = φ + 1 ↓     ↓     ↓ φ 3 = φ 2 + φ

The pattern continues! Here is a short list:

Note how each power is the two powers before it added together! The same idea behind the Fibonacci Sequence (see below).

Calculating It

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

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

• A) divide 1 by your value (=1/value)
• C) now use that value and start again at A

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:

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 = 3.14159265... 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:

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

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:

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

Or search by topic

Number and algebra

• The Number System and Place Value
• Calculations and Numerical Methods
• Fractions, Decimals, Percentages, Ratio and Proportion
• Properties of Numbers
• Patterns, Sequences and Structure
• Algebraic expressions, equations and formulae
• Coordinates, Functions and Graphs

Geometry and measure

• Angles, Polygons, and Geometrical Proof
• 3D Geometry, Shape and Space
• Measuring and calculating with units
• Transformations and constructions
• Pythagoras and Trigonometry
• Vectors and Matrices

Probability and statistics

• Handling, Processing and Representing Data
• Probability

Working mathematically

• Thinking mathematically
• Mathematical mindsets
• Cross-curricular contexts
• Physical and digital manipulatives

For younger learners

• Early Years Foundation Stage

Advanced mathematics

• Decision Mathematics and Combinatorics
• Advanced Probability and Statistics

The Golden Ratio

• Try this next
• Think higher
• Read: mathematics
• Explore further

and the human body

This exercise is divided into 3 parts:

A. The golden ratio

Distance from the ground to your belly button

Distance from your belly button to the top of your head

Distance from the ground to your knees

Distances A, B and C

Length of your hand

Distance from your wrist to your elbow

Now calculate the following ratios:

Distance from the ground to your belly button / Distance from your belly button to the top of your head

Distance from the ground to your belly button / Distance from the ground to your knees

Distance C / Distance B

Distance B / Distance A

Distance from your wrist to your elbow / Length of your hand

Write all your results on the following table:

Can you see anything special about these ratios?

B. The fibonacci sequence

Now look at the following sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

The following number is the sum of the previous two. This is Fibonacci's sequence.

Now do the following ratios on a calculator and give answers in non-fraction numbers:

As you go on and on dividing a number in the sequence by the previous number you get closer and closer to the number you discovered in the first part of the exercise, phi = $\phi$ = 1.6180339887498948482.

C. The golden rectangle

We can also draw a rectangle with the fibonacci number's ratio. From this rectangle we can then derive interesting shapes.

First colour in two 1x1 squares on a piece of squared paper:

Then draw a 2x2 square on top of this one:

Then draw a 3x3 square to the right of these:

Then draw a 5x5 square under these:

Then draw a 8x8 square to the left of these:

Then draw a 13x13 square on top of these:

We could go on like this forever, making bigger and bigger rectangles in which the ratio of length/ width gets closer and closer to the Fibonacci number.

Then place the compass tip on the bottom left corner of the 2x2 square and draw an arc like this:

Then place the compass tip on the left hand, top corner of the 3x3 square and do the same:

Do the same for the other three squares to obtain:

This shape is widely found in nature, can you find any other examples?

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.

High school geometry

Course: high school geometry   >   unit 4, geometry word problem: the golden ratio.

• Geometry word problem: Earth & Moon radii
• Geometry word problem: a perfect pool shot
• Similarity FAQ

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

Video transcript

• Math Article

Golden Ratio

In Mathematics, golden ratio – also known as golden mean, golden section, divine proportion – is a special number, which is often represented using the symbol “ϕ” (phi). The golden ratio finds its application in various fields such as arts, architecture, geometry, and so on. In this article, we are going to learn what the golden ratio is, the golden ratio formula, derivation and how the golden ratio is related to the Fibonacci sequence , in detail.

Table of Contents:

• What is the Golden Ratio?

Relation between Golden Ratio and Fibonacci Sequence

Practice question, golden ratio definition.

Two quantities are said to be in golden ratio, if their ratio is equal to the ratio of their sum to the larger of the two quantities. The golden ratio is approximately equal to 1.618. For example, if “a” and “b” are two quantities with a>b>0, the golden ratio is algebraically expressed as follow:

The golden ratio is an irrational number , which is the solution to the quadratic equation x 2 -x-1=0.

Some other terms that represent golden ratio include extreme and mean ratio, divine section, medial section, golden cut, and so on.

For example, divide the line into two sections. The two sections are in golden ratio if the ratio of the length of the larger section (say, “a”) to the length of the smaller section, (say, “b”) is equal to the ratio of their sum “a + b” to the larger section “a”.

Golden Ratio Formula

The golden ratio formula is used to calculate the value of the golden ratio.

From the definition of golden ratio, \(\begin{array}{l}\frac{a}{b} = \frac{a+b}{a}=\phi\end{array} \) , we get two equations.

i.e. a/b = ϕ …(1)

(a+b)/a = ϕ …(2)

Equation (2) can be written as:

(a/a) + (b/a) = ϕ

Therefore, the golden ratio formula is given by:

Golden Ratio Value Derivation

To derive the golden ratio value, multiply ϕ on both sides of equation (3), we get

Rearrange the above equation, we get

ϕ 2 -ϕ – 1 = 0, which is the form of quadratic equation .

Use the quadratic formula, \(\begin{array}{l}x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\end{array} \) .

Here, x = ϕ, a = 1, b=-1, c = -1

Hence, the two solutions obtained are:

ϕ = 1.618033.. and ϕ = -0.618033…

As ϕ is the ratio between two positive quantities, the value of ϕ should be the positive one.

Hence, the value of golden ratio ϕ is approximately equal 1.618 .

We know that the Fibonacci sequence is a special type of sequence in which each term in the sequence is obtained by adding the sum of two previous terms. Let us take the first two terms 0 and 1, then the third term is obtained by adding 0 and 1, which is equal to 1. The fourth term is found by adding the second term and third term (i.e. 1+1 = 2), and so on.

Hence, the Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21,..

There exists a special relation between the Fibonacci sequence and the golden ratio.

If we take two successive terms in the Fibonacci sequence, their ratio is very close to the golden ratio. If we take the bigger pair of Fibonacci numbers, the approximation is very close to the golden ratio.

Now, let us start with the term 2 in the Fibonacci sequence.

1. Which of the following represent the golden ratio formula?

• a/b = (a+b)/b
• a/b = (a+b)/a
• a/b = (a-b)/b
• a/b = (a-b)/a

2. The golden ratio ϕ is equal to

• ϕ – 1
• 1 – (1/ϕ)

To learn more Maths-related concepts easily, download BYJU’S – The Learning App, and explore many interesting videos.

Frequently Asked Questions about Golden Ratio

What is the golden ratio.

In Mathematics, two quantities are said to be in golden ratio, if their ratio is equal to the ratio of their sum to the larger of the two quantities.

Which symbol is used to represent the golden ratio?

The symbol used to represent golden ratio is ϕ (phi).

What is the value of the golden ratio?

The value of the golden ratio is approximately equal to 1.618.

How is the golden ratio related to the fibonacci sequence?

There exists a relation between the golden ratio and Fibonacci sequence, such that the ratio of two successive terms in the Fibonacci sequence is very close to the golden ratio.

Is the divine proportion the same as the golden ratio?

Yes, the divine proportion is the same as the golden ratio. The golden ratio is often represented using the terms, such as divine proportion, golden mean, golden proportion, golden section and so on.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

• Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

The Golden Ratio Examples

Tired of ads, cite this source, logging out…, logging out....

You've been inactive for a while, logging you out in a few seconds...

W hy's T his F unny?

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

10.4: Fibonacci Numbers and the Golden Ratio

• Last updated
• Save as PDF
• Page ID 32003

• Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier
• Coconino Community College

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

A famous and important sequence is the Fibonacci sequence, named after the Italian mathematician known as Leonardo Pisano, whose nickname was Fibonacci, and who lived from 1170 to 1230. This sequence is:

\[\{1,1,2,3,5,8,13,21,34,55, \ldots \ldots \ldots\} \nonumber \]

In other words, to get the next term in the sequence, add the two previous terms.

\[\{1,1,2,3,5,8,13,21,34,55,55+34=89,89+55=144, \cdots\} \nonumber \]

The notation that we will use to represent the Fibonacci sequence is as follows:

\[f_{1}=1, f_{2}=1, f_{3}=2, f_{4}=3, f_{5}=5, f_{6}=8, f_{7}=13, f_{8}=21, f_{9}=34, f_{10}=55, f_{11}=89, f_{12}=144, \ldots \nonumber \]

Example \(\PageIndex{1}\): Finding Fibonacci Numbers Recursively

Find the 13th, 14th, and 15th Fibonacci numbers using the above recursive definition for the Fibonacci sequence.

First, notice that there are already 12 Fibonacci numbers listed above, so to find the next three Fibonacci numbers, we simply add the two previous terms to get the next term as the definition states.

Therefore, the 13th, 14th, and 15th Fibonacci numbers are 233, 377, and 610 respectively.

Calculating terms of the Fibonacci sequence can be tedious when using the recursive formula, especially when finding terms with a large n. Luckily, a mathematician named Leonhard Euler discovered a formula for calculating any Fibonacci number. This formula was lost for about 100 years and was rediscovered by another mathematician named Jacques Binet. The original formula, known as Binet’s formula, is below.

A somewhat more user-friendly, simplified version of Binet’s formula is sometimes used instead of the one above.

All around us we can find the Fibonacci numbers in nature. The number of branches on some trees or the number of petals of some daisies are often Fibonacci numbers

Figure \(\PageIndex{4}\): Fibonacci Numbers and Daisies

a. Daisy with 13 petals b. Daisy with 21 petals

(Daisies, n.d.)

Fibonacci numbers also appear in spiral growth patterns such as the number of spirals on a cactus or in sunflowers seed beds.

Figure \(\PageIndex{5}\): Fibonacci Numbers and Spiral Growth

a. Cactus with 13 clockwise spirals b. Sunflower with 34 clockwise spirals and 55 counterclockwise spirals

(Cactus, n.d.) (Sunflower, n.d.)

Another interesting fact arises when looking at the ratios of consecutive Fibonacci numbers.

Example \(\PageIndex{5}\): Powers of the Golden Ratio

Helping with Math

Golden Ratio

Introduction.

Ratios hold a prominent place in the world of mathematics. We use different ratios in everyday life for the purpose of carrying out various measurements. One such ratio is the golden ratio. What is golden ratio and what is its significance? Let us find out.

The golden ratio is the ratio of two numbers such that their ratio is equal to the ratio of their sum to the larger of the two quantities. In other words, two quantities are said to be in golden ratio, if their ratio is equal to the ratio of their sum to the larger of the two quantities. The Golden Ratio was first discovered in the 1500s and was called “The Divine Proportion” in a book of the same title by Luca Pacioli. Therefore, the golden ratio is also known as the golden mean, golden section or divine proportion. It is also believed that the first person to officially name and define the Golden Ratio was the mathematician Euclid, in his treatise ‘Elements’ (written around 300BC), although the earlier mathematician Hippasus was the first to identify that the Golden Ratio wasn’t a whole number.

Symbol for Golden Ratio

• The golden ratio is often represented using the symbol “ϕ” (phi).

Approximate Value of Golden Ratio

• The approximate value of the golden ratio is 16.18.

Let us understand the golden ratio through an example.

Let “ a “ and “ b” be two quantities such that a and b are both positive numbers, i.e. a > b > 0, then the golden ratio of and b will be represented as – 

$\frac{a}{b} = \frac{a+b}{a}$ =   ϕ

Solving the above equation, we get,

a 2 – ab – b 2 = 0

The solution, ϕ to the above equation will be an irrational number such that

ϕ = $\frac{-1+ \sqrt{5}}{2}$ = 1.618033988 . . . . . . . . . . . . . . 

Consider the above figure. In this figure, we can see line segment AB and line segment BC. The ratio of the length of AB to AC is ϕ, which is the Golden Ratio. However, the ratio of segment BC to the whole segment AC is also ϕ.

Why Golden Ratio is considered so important?

The golden ratio is not just a factor obtained for a quadratic equation that has an irrational number as a solution. It is much more than this. The golden ratio is considered important for designing purposes with even the great artists such as Leonardo a Vinci having used the golden ratio in his art work. It is also believed that using the golden ratio can be useful to create the most pleasing and visually attractive shapes even the Parthenon in Athens, Greece is believed to have been built on the calculations based on the golden ratio. Some other examples include The Farnsworth House in the USA, The Great Stupa at Borobudur in Indonesia and The assorted works of Piet Mondrian. Not just this, one can simply create a simple design using a golden ratio. 

Formula for Golden Ratio

Let us now discuss the formula for the golden ratio.

We have learnt above that if “ a “ and “ b” are two quantities such that a and b are both positive numbers , i.e. a > b > 0, then the golden ratio of and b will be represented as – 

$\frac{a}{b} = \frac{a+b}{a}$ = ϕ . . . . . . . . . . . . .  . . . . . .  ( 1 )

Now, from the above equation, two more equations can be derived. These equations are – 

$\frac{a}{b}$ = ϕ . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

$\frac{a+b}{a}$ = ϕ . . . . . . . . . . . . . . . . . . . . . . . . 3 )

If we carefully observe the equation 2, we can see that this equation can also be written as  – 

$\frac{a}{a} + \frac{b}{a}$ = ϕ

This can equation can further be simplified to be written as – 

1 + $\frac{b}{a}$ = ϕ . . . . . . . . . . . . . . . . . . . . . . . ( 4 )

Now, if we observe the equation ( 1 ), we can say that –

$\frac{a}{b}$ = ϕ therefore ,

$\frac{b}{a} = \frac{1}{ϕ}$ . . . . . . . . . . . . . . . . . . . . . . . .  . . . . .  ( 5 )

Substituting the above values obtained in the equation ( 4 ), we get

1 + $\frac{1}{ϕ}$ = ϕ which is the formula for the golden ratio.

Hence, the formula for the golden ratio is given by 1 + $\frac{1}{ϕ}$ = ϕ

Let us now leant how the value of the golden ratio has been derived.

Derivation for the Golden Ratio Value

We have learnt above that the formula for the golden ratio is given by

1 + $\frac{1}{ϕ}$ = ϕ . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

⇒ ϕ + 1 = ϕ 2

The above equation can be rearranged to written as – 

ϕ 2 – ϕ – 1 = 0 . . . . . . . . . . . . . . . . . . . ( 2 )

Now, the above equation is a quadratic equation of the form a x 2 + b x  + c = 0 

We know that in order to solve  quadratic equation, we use the following formula – 

x = $\frac{-b±\sqrt{b^2-4ac}}{2 a}$ . . . . . . . . . . . . . . . . . . . ( 3 ) 

From equation ( 1 ) we have,

x = ϕ, a = 1, b = – 1 and c = – 1

Substituting these values in the equation ( 3 ) we get,

ϕ = $\frac{- (-1) ± \sqrt{( -1 )^2-4 x (1) x ( -1 )}}{2 x ( 1 )} = \frac{1± \sqrt{1+4}}{2} = \frac{1± \sqrt{5}}{2}$

Thus, two solutions that can be obtained by solving the above equation are – 

ϕ = $\frac{1 + \sqrt{5}}{2}$ . . . . . . . . . . . . . . .  ( 4 ) and

ϕ = $\frac{1 – \sqrt{5}}{2}$ . . . . . . . . . . . . . . .  ( 5 )

Simplifying the equations ( 4 ) and ( 5 ) we will get

ϕ = 1.618033 . . . . . . . . . .  and

ϕ = – 1.618033 . . . . . . . . . .

Now, it is important to recall here that by the basic definition of the golden ratio is has to a ratio between two positive values. This means that the golden ratio cannot be a negative number. Hence, we shall discard the negative value of that we have obtained above. Thus we are left with one value of which is 1.618033 . . . . . . . . . .

Hence, the golden ratio is approximately equal to 1.618033 . . . . . . . . . . . . . . . 

Let us now recall another series that is widely used in mathematics, i.e. Fibonacci series. Is there any relation between the golden ratio and the Fibonacci series? Let us find out.

Relation between Golden ratio and Fibonacci Series

Let us first recall what we mean by the Fibonacci series. We know that the Fibonacci sequence is a special type of sequence in which each term in the sequence is obtained by adding the sum of two previous terms.  The Fibonacci series was named after an Italian mathematician named Leonardo Pisano Bogollo, who was later known as Fibonacci. The Fibonacci series is the sequence of numbers which are also known as Fibonacci numbers , where every number is sum of the preceding two numbers, such that the first two terms are ‘ 0 ‘ and ‘ 1 ‘. The formula to find the Fibonacci series can be given to express the ( n + 1 ) t h term in the sequence is defined using the recursive formula, such that F o = 0, F 1 = 1, to give F n . For example, let us take the first two terms 0 and 1, then the third term is obtained by adding 0 and 1, which is equal to 1. The fourth term is found by adding the second term and third term ( i.e. 1 + 1 = 2 ), and so on. Hence, the Fibonacci sequence is 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , . . . . . . . . . . . . . .

Can we say that there exists a relation between the golden ratio and the Fibonacci series? Yes, there does exist a special relation between the golden ratio and the Fibonacci series. What is this relation? Let us find out.

Let us consider two successive terms of the Fibonacci series. If we carefully observe the ratio of these two terms, we will find that this ratio is very close to the golden ratio. If we take two large numbers which are successive terms of the Fibonacci series, we will find that this ratio is very close to the golden ratio. For better understanding, let us place the successive values of the Fibonacci series in the form of a table and find their ratios. We will have,

From the above values we can see that the ratio of successive terms of the Fibonacci series is very close to the golden ratio.

Golden Rectangle

Similar to the golden ratio, we have a golden rectangle as well. Let us consider a rectangle assuming that its sides satisfy the golden ratio. Such a rectangle is called a golden rectangle. We then add a line segment in the rectangle that separates the rectangle into a perfect square while creating another rectangle as well. This news rectangle is the golden rectangle. Thus, a golden rectangle again consists of a square and a smaller rectangle, which is itself a golden rectangle.

If we repeat this process and draw a curve through the successive squares, we obtain a golden spiral. The spiral thus obtained is a spiral whose logarithmic growth curve is  . now let us consider another geometric figure which is known as a regular pentagon. We know that a regular pentagon is a shape that is made of 5 edges. Suppose we have a regular pentagon having a side “ a “ . We then draw diagonals within the pentagon. The length of these diagonals will be a . .  

Real Life Applications of Golden Ratio

The golden ratio is not just a standalone formula to be used for mathematical calculations. It has been widely used in real life situations as well. Some of these areas where we find the application of the golden ratio are – 

• The golden ratio is found in different arts, architecture and designs. 
• The designs of famous structures such as The Great Pyramid of Egypt and the Parthenon have either been partially or completely based on the concept of the golden ratio.
• In the 1500s the golden ratio was also known as the “ Divine Proportion “.
• Great artists like Leonardo Da Vinci used the golden ratio in a few of his masterpieces.
• Even the human face has proportions that are approximately the Golden Ratio, so beauty experts rely on it to create aesthetically pleasing looks.
• The standard widescreen aspect ratio for televisions and monitors is about 1.7, which is pretty close to the golden ratio.
• Many houses have rooms whose dimensions obey the Golden Ratio.
• In modern mathematics, the golden ratio occurs in the description of fractals, figures that exhibit self- similarity and play an important role in the study of chaos and dynamical systems.

Golden Ratio in Human Body

It is not only the objects and monuments where we find the use as well as applications of the golden ratio. The golden ratio can be observed in the human body as well. In fact, the centre of many proportions in the human body happens to be the golden ratio. The shape of the human face as well as the ratio of the height of the navel to the height of the body is all based on the golden ratio. There are many possible ratios of the human body, with the ideal one being assumed the one that obeys the golden ratio. 

Key Facts and Summary

• The golden ratio is the ratio of two numbers such that their ratio is equal to the ratio of their sum to the larger of the two quantities.
• Let “ a “ and “ b” be two quantities such that a and b are both positive numbers, i.e. a > b > 0, then the golden ratio of and b will be represented as – $\frac{a}{b} = \frac{a+b}{a}$ = ϕ.
• The formula for the golden ratio is given by 1 + $\frac{1}{ϕ}$ = ϕ.
• The ratio of successive terms of the Fibonacci series is very close to the golden ratio.
• Similar to the golden ratio, we have a golden rectangle as well. Let us consider a rectangle assuming that its sides satisfy the golden ratio. Such a rectangle is called a golden rectangle.
• The golden ratio is not just a stand alone formula to be used for mathematical calculations. It has been widely used in real life situations as well.
• Great artists like Leonardo Da Vinci are believed to have used the golden ratio in a few of his masterpieces.

Recommended Worksheets

Ratio and Proportion (Armistice Day Themed) Math Worksheets Understanding Ratio between Two Quantities 6th Grade Math Worksheets Solving Proportional Relationships Between Two Quantities 7th Grade Math Worksheets

Link/Reference Us

We spend a lot of time researching and compiling the information on this site. If you find this useful in your research, please use the tool below to properly link to or reference Helping with Math as the source. We appreciate your support!

<a href="https://helpingwithmath.com/golden-ratio/">Golden Ratio</a>

"Golden Ratio". Helping with Math . Accessed on May 14, 2024. https://helpingwithmath.com/golden-ratio/.

"Golden Ratio". Helping with Math , https://helpingwithmath.com/golden-ratio/. Accessed 14 May, 2024.

Additional Number Sense Theory:

Latest worksheets.

The worksheets below are the mostly recently added to the site.

Classifying Shapes by Lines and Angles 4th Grade Math Worksheets

Understanding Number and Shape Patterns 4th Grade Math Worksheets

Prime and Composite Numbers 4th Grade Math Worksheets

Comparing Multi-digit Numbers 4th Grade Math Worksheets

Solving Word Problems Involving Perimeter and Area of Rectangle 4th Grade Math Worksheets

Understanding Decimal Notations 4th Grade Math Worksheets

Multiplicative Comparisons and Equations 4th Grade Math Worksheets

Writing Multi-Digit Numbers Using Base 10 Numerals 4th Grade Math Worksheets

Understanding Angles and Its Measures 4th Grade Math Worksheets

Kindergarten Geometry Free Bundle

Golden Ratio

Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

There are several different ways to express the golden ratio. 4 different ways are listed below.

\(\displaystyle \text{golden ratio } \displaystyle\phi = \frac{1+\sqrt{5}}{2}\), \(\displaystyle \text{golden ratio } \displaystyle\phi = 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+…\frac{1}{1+\frac{1}{1+…}}}}}\), \(\displaystyle \text{golden ratio } \displaystyle\phi = \frac{a}{b} \text{ such that } \frac{a+b}{a}=\frac{a}{b}\), \(\displaystyle \text{golden ratio } \displaystyle\phi \approx 1.61803398875 \), see related pages\(\), \(\bullet\text{ fibonacci sequence}\) \(\,\,\,\,\,\,\,\,0,1,1,2,3,5,8,13,21…\), \(\bullet\text{ golden ratio}\) \(\,\,\,\,\,\,\,\,1.61803398875…\), \(\bullet\text{ pascal’s triangle}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ binomial theorem}\) \(\,\,\,\,\,\,\,\,\displaystyle\frac{n}{(n-(k-1))(k-1)}a^{(n-(k-1)))b^{k-1}}\), \(\bullet\text{ andymath homepage}\).

The Golden Ratio, also known as the Divine Proportion or Phi, is a mathematical concept that has been used in art, architecture, and design for centuries. It is a ratio of approximately 1:1.618, and it is believed to be aesthetically pleasing to the human eye. The Golden Ratio is often defined as the ratio of the longer side of a rectangle to the shorter side, when the ratio of the sum of the sides to the longer side is equal to the ratio of the longer side to the shorter side. In other words, it is the ratio of two quantities that is the same as the ratio of the sum of those quantities to the larger one. People often study the Golden Ratio because it has been used in the design of many famous works of art and architecture, such as the Pyramids of Giza and the Parthenon. It is also believed to be present in natural forms, such as seashells and pinecones, which may contribute to its perceived beauty. The Golden Ratio is typically covered in geometry or advanced math classes, as it involves concepts such as ratios, proportions, and irrational numbers. It may also be discussed in art or design classes as it relates to aesthetics. One common mistake when working with the Golden Ratio is confusing it with the Golden Rectangle, which is a rectangle with sides that are in the ratio of the Golden Ratio. It is important to remember that the Golden Ratio is a ratio and the Golden Rectangle is a geometric shape. A fun fact about the Golden Ratio is that it has been used for centuries, with some historians tracing its use back to ancient Greece. It was also referred to by Leonardo da Vinci as the "Divine Proportion" and was a key concept in the work of artists such as Salvador Dali and Le Corbusier. The Golden Ratio was first formally described by the Greek mathematician Euclid, but it is likely that it was known and used by earlier civilizations. Euclid's work on the Golden Ratio, as well as the work of other ancient mathematicians, has had a lasting impact on the fields of mathematics and art. Some related topics to the Golden Ratio include Fibonacci numbers, geometry, and the golden spiral. The Golden Ratio is also often compared to the concept of balance or harmony, as it is believed to create aesthetically pleasing compositions. 5 real world examples of Golden Ratio The Parthenon in Athens, Greece is considered a prime example of the use of the Golden Ratio in architecture. The width of the temple is in the Golden Ratio to its length, and the columns and other features of the temple also exhibit the Golden Ratio. The Mona Lisa, a famous painting by Leonardo da Vinci, is believed to use the Golden Ratio in its composition. The dimensions of the painting and the placement of elements within it are thought to conform to the Golden Ratio. The human face is often said to embody the Golden Ratio, with the distance between the eyes and the distance between the eyes and the mouth both being in the Golden Ratio to the length of the face. The nautilus shell exhibits the Golden Ratio in its spiral shape. The growth of the shell follows a logarithmic spiral, which conforms to the Golden Ratio. The spiral galaxy M51, also known as the Whirlpool Galaxy, exhibits the Golden Ratio in its structure. The arms of the galaxy swirl out from the center in a pattern that conforms to the Golden Ratio. 5 other math topics that use Golden Ratio Here are five other math topics that use the golden ratio: Geometry: The golden ratio appears in various geometric figures, such as the golden rectangle, which is a rectangle with sides in the ratio of the golden ratio, and the golden spiral, which is a logarithmic spiral that follows the golden ratio. Art: The golden ratio has been used by artists throughout history to create aesthetically pleasing compositions. It is believed that the golden ratio appears in many famous works of art, such as the Mona Lisa and the Parthenon. Architecture: The golden ratio has also been used in the design of buildings and other structures, such as the Great Pyramid of Giza and the Notre Dame cathedral. Music: Some musicians and composers have used the golden ratio in the composition of music, including the French composer Debussy and the American composer Phil Spector. Biology: The golden ratio has been observed in various natural phenomena, such as the arrangement of branches and leaves on plants, the shape of seashells, and the proportions of the human body. Some scientists believe that the golden ratio may play a role in the evolution of these and other natural forms.

About andymath.com, andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. if you have any requests for additional content, please contact andy at [email protected] . he will promptly add the content. topics cover elementary math , middle school , algebra , geometry , algebra 2/pre-calculus/trig , calculus and probability/statistics . in the future, i hope to add physics and linear algebra content. visit me on youtube , tiktok , instagram and facebook . andymath content has a unique approach to presenting mathematics. the clear explanations, strong visuals mixed with dry humor regularly get millions of views. we are open to collaborations of all types, please contact andy at [email protected] for all enquiries. to offer financial support, visit my patreon page. let’s help students understand the math way of thinking thank you for visiting. how exciting.

Fibonacci Numbers and Golden Ratio

Related Topics: More Lessons for Calculus Math Worksheets

The Fibonacci Sequence and the Golden Ratio Introduces the Fibonacci Sequence and explores its relationship to the Golden Ratio.

The Golden Ratio

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.