## The 27-63-90 Triangle

I’m not as pleased about finding this special right triangle as I was with the previous ones. For one thing, I didn’t use a regular polygon to derive it, but instead used the 36-54-90 triangle I had previously found, and applied a half-angle trigonometric identity to find the tangent of 27 degrees, as the tangent of half of 54 degrees. According to this identity, tan(27⁰) = (1 – cos(54⁰))/sin(54⁰). Once this gave me the leg lengths, I simply used the Pythagorean Theorem to determine the length of the hypotenuse. Finally, I checked all of this using decimal approximations.

The triple-nested radical in the expression for the hypotenuse is no cause for celebration, either. If anyone knows of a way to put this in simpler exact terms, please let me know.

I go by RobertLovesPi on-line, and am interested in many things, the majority of which are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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### 3 Responses to The 27-63-90 Triangle

1. Anonymous says:

how did you come up figure out that would work though?

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2. It’s derived from the 36-54-90 triangle, which is itself derived from the regular pentagon, and uses the fact that such pentagons’ diagonals and sides are in the golden ratio.

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3. Anonymous says:

The hypotenuse is equal to: 2*sqrt[5+sqrt(5)]-sqrt(2)*(sqrt(5)-1).

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