So I heard that if one inscribes the largest circle that can fit into a equilateral triangle, then divides the perimeter of the triangle by the diameter of the inscribed circle, it gives a value which can be called "triangle $\pi$", and that value ($\sqrt{27}$) can be used in the place of regular $\pi$ to derive volumes of the other platonic solids. Is that true? Is there a different $\pi$ for triangles? What is that value? Is it close to $\sqrt{27}$? Can it be used to find volumes of platonic solids, especially the icosahedron and the one that looks like a pyramid flipped and stacked on its twin? 4 part question. Thanks we have been arguing about it at work for weeks
Is there a value for $\pi$ that relates to triangles?
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$\begingroup$
euclidean-geometry
triangles
pi
platonic-solids
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5So, what is the question? – 2012-06-14
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2$\pi$ is a constant, it has only one value... – 2012-06-15
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0The word you are looking for is *octahedron*. – 2012-06-15
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1If you want to know the volumes of the Platonic solids, try http://en.wikipedia.org/wiki/Platonic_solid#Radii.2C_area.2C_and_volume. – 2012-06-15
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1The relationship of that number to triangles is somewhat analogous to the relationship of $\pi$ to circles, but calling it "triangle pi" is problematic in two respects: (1) It is not standard terminology, and (2) There are all sorts of special properties of the number $\pi$ that would not apply to that number. $\pi$, for example is a transcendental number. I can imagine it being called "triangle pi$ within the context of a particular article about it, but I'd be a bit surprised if the author proposed adopting that language as standard terminology. – 2012-06-15
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0I'm not sure if you get notified by edits to answers, so I'm leaving you a comment to say that I've added a little bit to my answer that you might find interesting. – 2012-06-15
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0It's actually very interesting. I've been working on just these sorts of questions to create some new geometry. Thanks! – 2015-09-28