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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

$\pi \approx \frac{4}{\sqrt{\phi}}$

The next post

Note by Bob Krueger
3 years, 6 months ago

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I didn't know that this triangle had a name! Learning something new everyday. Thanks! Staff · 3 years, 6 months ago

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. · 3 years, 6 months ago

It is the real information. Really nice, · 3 years, 6 months ago