# How to construct a golden rectangle

This example demonstrates how to construct the golden rectangle using a compass and ruler. The golden rectangle is a rectangle whose sides form the golden ratio which is a naturally occurring number approximately equal to .

## Steps

Start by drawing a square with a side length of 2. The length of 2 is chosen instead of a length of 1 because it will make the math easier later on.

Place a point in the middle of the bottom edge of the square so that the edge is divided into two equal parts.

Draw a line from the point to the top right corner of the square. This line forms the radius of the circle which we are about to draw. Using the Pythagorean’s Theorem we know that the length of the radius is . Also, extend the bottom edge of the square outwards in the right direction.

Using the point and the radius, draw the arc of the circle from the top right corner of the square in a clock-wise motion until the arc intersects with the bottom line we extended.

Draw a perpendicular line up from the point of intersection and another line which extends the top edge of the square to get the top right corner of the golden rectangle.

We have now finished constructing the golden rectangle. The ratio of the sides form the golden ratio represented by the greek letter (phi).

We can also scale the golden rectangle by a factor of two to get a normalized version of the golden rectangle where the shorter side’s length is 1.

The pythagorean theorem equates the square of the sides of a right triangle together.

The golden rectangle is a rectangle whose width divided by height is equal to the golden number (approximately 1.618).

The golden ratio is a number represented by the greek letter ϕ (phi). The value of ϕ is approximately 1.618 and is a naturally occurring number in nature. A golden rectangle is defined as the angle that has side lengths in ratio. Typically, a square must be constructed to build a golden rectangle. For those looking to draw a classic proportioned rectangle, you should consider drawing an angle first. Drawing a golden rectangle is an excellent way to find two unknown numbers that have a Golden Mean relationship with each other. Constructing a golden rectangle can be quite challenging. If you want to construct a golden rectangle then you will have to follow some simple instructions to learn this ancient Greek methodology.

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### Instructions

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Use a straightedge and a measuring device to draw a perfect square. You could choose a square of any size but make sure there is enough room at the top of your paper. It is important to keep some space at the top for the extension.

Now consider bisecting one side of the square using the compass method. This can be done by opening the compass to just over half of the length of the side of the square. Place the compass on one end of the baselines and draw arcs; one on below and one above the line. Put the point of the compass on the other end and repeat the process to make the two arcs intersect below and above the line. Now consider drawing a line through these intersecting points. Connecting line should pass through the square’s sides in the middle of the square.

Place the pencil one of the square’s corners on the opposite side and place the compass pointer on the halfway mark. Draw an arc large enough to make sure there is a point where the extension intersects the arc. As it joins the arc extend the bisected side of the square.

Repeat step 2 and step 3 for the other side of the baseline. When you are done with it, you should have extension lines of the same size on all four sides of the square. To construct a golden rectangle, consider joining the two extension lines. Erase all extra lines if you want to see only your golden rectangle.

The ratio, called the Golden Ratio, is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes. This rectangle, called the Golden Rectangle, appears in nature and is used by humans in both art and architecture. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births. The ratio is close to 1.618. Whoever first discovered these intriguing manifestations of geometry in nature must have been very excited about the discovery.
A study of the Golden Ratio provides an intereting setting for enrichement activities for older students. Ideas involved are: ratio, similarity, sequences, constructions, and other concepts of algebra and goemetry.

Finding the Golden Ratio. Consider a line segment of a length x+1 such that the ratio of the whole line segment x+1 to the longer segment x is the same as the ratio of the line segment, x, to the shorter segment, 1.

A positive root of this equation is

or 1.61803. This irrational number, or its reciprocal

is known as the Golden Ratio, phi .

Now we will construct the Golden Rectangle. First we will construct a square ABCD.

Now we will construct the midpoint E of DC.

Extend DC. With center E and radius EB, draw an arc crossing EC extended at C.

Construct a perpendicular to DF at F.

Extend AB to intersect the perpendicular at G.

AGFD is a Golden Rectangle.

Now we will measure the length and width of the rectangle. Then we will find the ratio of the length to the width. This should be close to the Golden Ratio (approximately 1.618).

Now we will take our Golden Rectangle and continue to divide it into other Golden Rectangles.

Within this one large Golden Rectangle there are six other Golden Rectangles.
When you measure each Golden Rectangles length and width you will see that the ratio of the length to the width is the Golden Ratio (spproximately 1.618).

Now we will construct the spiral through the whole Golden Rectangle.

If you’re a designer or artist, you should know about the Golden Ratio. We explain what it is and how you can use it.

Creating a Golden Rectangle is pretty straightforward, and starts with a basic square. Follow the steps below to create your own Golden Ratio:

### 01. Draw a square

Start by drawing a square of any size. The side of this will form the length of the ‘short side’ of the rectangle.

### 02. Divide the square

Next, divide your square in half with a vertical line down the centre. This will leave you with two rectangles.

### 03. Draw a diagonal line

In one of these rectangles, draw a straight line from one corner to the opposite corner.

### 04. Rotate the line

Rotate this line, pivoting from the bottom (or top) point, until it lines up with the bottom of the first rectangle.

### 05. Create a new rectangle

Create a rectangle using the new horizontal line and original rectangle as guides. That’s your Golden Rectangle.

Next page: How to use the Golden Ratio in your design work

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The ratio, called the Golden Ratio, is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes. This rectangle, called the Golden Rectangle, appears in nature and is used by humans in both art and architecture. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births. The ratio is close to 1.618.

Look at the rectangle below. Do you think the ratio of length to width is golden? To find out for sure click on the rectangle above and explore with the rectangle that appears.

• Drag point E. Describe what happens?
• Drag point B. Describe what happens?

Now we will construct the Golden Rectangle. First we will construct a square ABCD. Now we will construct the midpoint E of DC. Extend DC. With center E and radius EB, draw an arc crossing EC extended at C. Construct a perpendicular to DF at F. Extend AB to intersect the perpendicular at G. AGFD is a Golden Rectangle.

Now we will measure the length and width of the rectangle. Then we will find the ratio of the length to the width. This should be close to the Golden Ratio (approximately 1.618). Now we will take our Golden Rectangle and continue to divide it into other Golden Rectangles. Within this one large Golden Rectangle there are six other Golden Rectangles.
When you measure each Golden Rectangles length and width you will see that the ratio of the length to the width is the Golden Ratio (approximately 1.618). Now we will construct the spiral through the whole Golden Rectangle.

There’s a mathematical ratio commonly found in nature—the ratio of 1 to 1.618—that has many names. Most often we call it the Golden Section, Golden Ratio, or Golden Mean, but it’s also occasionally referred to as the Golden Number, Divine Proportion, Golden Proportion, Fibonacci Number, and Phi.

You’ll usually find the golden ratio depicted as a single large rectangle formed by a square and another rectangle. What’s unique about this is that you can repeat the sequence infinitely and perfectly within each section.

Quick announcement – EmptyEasel has created a quicker, easier way for artists to have their own art website. Click here to learn more and get a simple art website of your own! If you take away the big square on the left, what remains is yet another golden rectangle. . . and so on.

### The golden ratio in art and architecture

The appearance of this ratio in music, in patterns of human behavior, even in the proportion of the human body, all point to its universality as a principle of good structure and design.

Used in art, the golden ratio is the most mysterious of all compositional strategies. We know that by creating images based on this rectangle our art will be more likely to appeal to the human eye, but we don’t know why.

Some scholars argue that the Egyptians applied the golden ratio when building the great pyramids, as far back as 3000 B.C.

In 300 B.C. Euclid described the golden section in his writing of Euclid’s Elements, and before that, around 500 B.C., Pythagoras claimed that the golden ratio is the basis for the proportions of the human figure.

The ancient Greeks also used the golden ratio when building the Parthenon. Artists throughout history, like Botticelli and Leonardo daVinci, have used the golden rectangle, or variations of it, as the basis for their compositions.

Here’s da Vinci’s painting, The Last Supper, with golden sections highlighted. Golden rectangles are still the most visually pleasing rectangles known, and although they’re based on a mathematical ratio, you won’t need an iota of math to create one.

### How to make a rectangle based on the golden ratio

If you want to use a golden rectangle in your own compositions, here’s how you can make that happen without any special tools or mathematical formulas.

1. Begin with a square, which will be the length of the short side of the rectangle.

2. Then draw a line that divides it in half (forming two rectangles).

3. Draw a line going from corner to opposing corner of one of those halves. 4. Rotate the top point of that diagonal line downward until it extends your square.

5. Finish off the rectangle using that diagonal length as a guide for the long side of your golden rectangle. It’s that simple.

### Visual points of interest inside a golden rectangle

Any square or rectangle (but especially those based on the golden ratio) contain areas inside it that appeal to us visually as well. Here’s how you find those points:

1. Draw a straight from each bottom corner to its opposite top corner on either side. They will cross in the exact center of the format.

2. From the center to each corner, locate the midway point to each opposing corner. These points—represented by the green dots in the diagram above—are called the “eyes of the rectangle.”

### How to use the “eyes” of a golden rectangle

One strategy often used by artists is to locate focal points or areas of emphasis around and within these eyes, creating a strong visual path in their compositions.

Edward Hopper’s composition, below, sets the sailboat right on the lower right eye (with the tip of the sails extending nearly to the upper right eye). In this painting, Carolyn Anderson places her subject’s hands around that spot too. J.M.W. Turner uses the angle of his waves to create an arch that circles through the lower right and lower left eyes. Notice that the focus of the scene is then captured within all four eyes, too.

How you use the golden ratio or these “eyes” to direct a viewers’ visual path is limited only by your imagination. Try using the eyes of the rectangle in your next painting and see what difference it can make in the strength of your composition.

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Starting with a square, construct a golden rectangle. How do you obtain a golden spiral?

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Today I will be discussing what the golden ratio is (otherwise known as the golden mean) and how we can use it to improve your artworks.

### What Is The Golden Ratio?

The golden ratio is the ratio of approximately 1 to 1.618. These are extremely important numbers to mathematicians. But what do they mean to us artists?

Well there have been studies which suggest designs set out using the golden ratio are aethetically pleasing. We can use the golden ratio to help design our paintings and position our subjects.

Who would have thought art and maths could have such a close connection? Luca Pacioli (a contemporary of Leonardo da Vinci) went as far as saying:

“Without mathematics there is no art.”

### History Of The Golden Ratio

The golden ratio has been around for some time and has influenced many areas of life, including architecture, maths, design and of course art.

Here is a rough timeline of the golden ratio’s history according to author Priya Hemenway:

• Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.
• Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English “extreme and mean ratio”.
• Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci. We will discuss the Fibonacci sequence later in this post.
• Luca Pacioli (1445–1517) defines the golden ratio as the “divine proportion” in his Divina Proportione.
• Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
• Martin Ohm (1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.
• Édouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.

### Calculations

I will try and keep this simple (as we do not need to understand all the complexities of the golden ratio as artists).

The golden ratio can be calculated as follows:

That weird symbol at the end represents the golden ratio.

I find this equation easier to understand in pictural format: So a+b is to a as a is to b.

Confused yet? Keep reading as it becomes easier to understand when we apply it to certain situations.

### The Golden Rectangle

Below is a golden rectangle, which means the side lengths are in golden ratio. If you take away that square on the left, another rectangle will remain which is also in golden ratio. This could continue indefinately.

There is some kind of peacefulness and beauty in infinite numbers, which is possibily why the golden ratio is so popular in design. Creating the golden rectangle is easy using these steps. All you need is a compass.

Step 1 – Construct a simple square.

Step 2 – Draw a line down the middle of the square. Step 3 – Grab your compass and place one point at the intersection at the bottom middle and draw down from the edge of top right corner, as shown below. Step 4 – Complete the golden rectangle. Note: This is for demonstration purposes only so it may not be the exact proportions of the golden ratio.

### The Fibonacci Sequence

The following is 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.

When we take any two successive (one after the other) in the sequence, their ratio is very close to the golden ratio.

In fact, the later the numbers are in the sequence, the closer it becomes to the golden ratio.

This relationship between the Fibonacci sequence and the golden ratio is shown below: