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The Golden Rectangle: Definition, Formula & Examples

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  • 0:00 Fibonacci Spiral &…
  • 1:11 1. The Sides of a…
  • 3:20 2. The Golden Ratio
  • 4:40 3. Creating a Golden Rectangle
  • 5:30 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will learn what a golden rectangle is and the relationship between the lengths of its sides. We will also look at the relationship between these types of rectangles, the Fibonacci spiral, and the golden ratio.

The Fibonacci Spiral & Golden Rectangle

Have you ever looked at an interactive weather map showing hurricane activity? If you have, you have probably observed that a hurricane takes on a spiral shape. You may think this shape looks a bit like a seashell or a winding staircase. Not only are you correct that these thing have the same shape, you are actually observing what is called the Fibonacci spiral. The Fibonacci spiral shows up in many areas of nature, art, architecture, astronomy, oceanography, and many other places. You may wonder where this Fibonacci spiral comes from, and the answer to that lies within the golden rectangle.

Do you have a rectangle size that is your favorite? That may sound like an odd question. However, in mathematics there is a certain type of rectangle that is found to be the most aesthetically pleasing to the eye. This rectangle is called the golden rectangle.

The Golden Rectangle with the Fibonacci Spiral
golden rectangle

The golden rectangle is a fascinating mathematical phenomenon. The rectangle possesses many properties, and holds many different patterns within. To fully understand what the golden rectangle is, we are going to look at three of its defining characteristics.

1: The Sides of a Golden Rectangle

A golden rectangle has the property that if its sides have lengths a and b, where a is the longer side, then the ratio of a to b is equal to the ratio of b to a - b. That is, the ratio of the length of the longer side to the length of the shorter side is equal to the ratio of the length of the shorter side to the length of the longer side minus the shorter side. In equation form, we have a / b = b / (a - b).

We can use this formula to analyze the golden rectangle. For example, suppose I have a golden rectangle, and I know that its shorter side has length 8 centimeters. I can plug 8 in for b in the formula, and then solve for a to find the length of the other side of the rectangle.

We get a / 8 = 8 / (a - 8), then we cross multiply, or multiply both sides by -8. That gives us a^2 - 8a = 64. Next, we subtract 64 from both sides to get a^2 - 8a - 64 = 0. Lastly, we solve for a using the quadratic formula, which is x = (-b+/- sqrt(b^2 - 4ac)) / 2a. Using our values, we have a = (8 +/- sqrt(8^2 - 4 * 1 * -64)) / 2 * 1. Then we simplify it to a = (8 +/- sqrt(64 + 256)) / 2, which becomes a = 12.944 or a = -4.944.

Since a is a length, it must be positive, so we see that the length of the longer side of the rectangle would be approximately 12.944 centimeters. We can also use this formula to create a golden rectangle or verify that a rectangle is a golden rectangle.

2: The Golden Ratio

The golden ratio is a special number in mathematics that has approximate value of 1.618. The exact value of the golden ratio is (sqrt(5) + 1) / 2. The golden ratio and the golden rectangle are intimately connected. This is because the ratio of the longer side of a golden rectangle to the shorter side is equal to the golden ratio.

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