What is the Golden Ratio in Math? - Definition & Examples

Lesson Transcript
Instructor: Emily Cadic

Emily has a master's degree in engineering and currently teaches middle and high school science.

The golden ratio in mathematics is used as a reference for creating patterns and shapes. Learn about the definition of the golden ratio and see some examples of it present in the world. Updated: 10/16/2021

Definition of the Golden Ratio

The golden ratio is an irrational number represented by the Greek letter phi (φ) that's used to create geometries with what many people consider the most eye-pleasing proportions. Some of the different formats used to express the golden ratio are shown here:

Golden ratio examples

The use of the golden ratio surged in popularity during the Renaissance, identifiable in both the art and architecture of the era. Some have also contended that the ancient Greeks intentionally constructed the Parthenon according to the golden ratio. In any case, phi continues to captivate our interest to this day, but how exactly does it work in action?

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  • 0:03 Definition of the Golden Ratio
  • 0:42 Golden Ratio and Geometry
  • 2:26 The Golden Ratio and…
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The Golden Ratio and Geometry

One of the simplest examples of the golden ratio in relation to geometry is a special line segment called the golden segment, illustrated here:

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In this segment, the ratio of the blue segment to the red segment is equal to the ratio of the red segment to the entire line from A to C. In other words, AB/BC = BC/AC.

But where does the 1.618 show up? If we set AB = 1, and BC = x, we will see an interesting result when we solve.

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Another notable appearance of the golden ratio in geometry comes in the form of the pentagram. Whether you believe it's a symbol of witchcraft or divinity, one thing is for certain: it's golden.

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Just like the line we saw earlier, the pentagram can be broken down into segments that are related through the golden ratio. If each side of the outer pentagon (or the green sides) is equal to 1 unit, then the length of each side of the inner pentagon (or the purple section) is 1/φ². Furthermore, the length of each of the five lines that create the perimeter of the star shape (or the orange part) would be equal to phi.

The presence of the golden ratio in geometry does not stop with the pentagram. The same relationships can be demonstrated in more complex three-dimensional shapes such as dodecahedrons and icosahedrons, which have 12 faces and 20 faces, respectively.

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