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The Golden Ratio: Kepler Triangle

Here is the previous post concerning the Golden Ratio. For a collection of all the posts concerning the Golden Ratio, click #GoldenRatio below.

I propose a question for you: If the smallest side of a right triangle is \(1\), and the sides are in geometric proportion, what are the other sides of the triangle? Try and answer this question yourself before reading further.

If the smallest side is \(1\), we can call the next leg \(r\) and the hypotenuse \(r^2\). By the Pythagorean Theorem, we know that \(1^2 + r^2 = r^4\). To make this equation look especially familiar, we will use the substitution \(a=r^2\). Then \(1 + a = a^2\). Thus \(a = r^2 = \phi\) and \(r = \sqrt{\phi}\) are the remaining two sides of the triangle. This is called a Kepler Triangle.

The Kepler Triangle has interesting properties. Half the base, height, and slant height of the great pyramid at Giza form a Kepler Triangle.

Next, let's construct a square with side length \(\sqrt{\phi}\) and the circumcircle of our Kepler Triangle. What are the perimeters of each? Well, the square has perimeter \(4\sqrt{\phi} \approx 5.088\) and the circumcircle (whose diameter is the hypotenuse of the Kepler Triangle) has circumference \(\pi \phi \approx 5.083\). These are strikingly similar! In fact, the error between them is less than a tenth of a percent! From this, we can get the approximation coincidence

\[\pi \approx \frac{4}{\sqrt{\phi}}\]

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Note by Bob Krueger
2 years, 9 months ago

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I didn't know that this triangle had a name! Learning something new everyday. Thanks! Calvin Lin Staff · 2 years, 9 months ago

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@Calvin Lin Kepler said that the two most interesting things from geometry were the Pythagorean Theorem and the golden ratio (from Wikipedia). So this special triangle is named after him. Bob Krueger · 2 years, 9 months ago

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It is the real information. Really nice, Muh. Amin Widyatama · 2 years, 9 months ago

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Woah Faraz Masroor · 2 years, 9 months ago

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