# Introducing the Piangle

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While trying to sleep last night I discovered a new triangle. I call it the Piangle.

What's a Piangle? Maybe this will make it clear.

The Piangle is an unraveled circle. Imagine cutting a radius, then draw some inner circles.

Next unroll each outline to the right.

This is a right triangle, so by the Pythagorean theorem the length of the hypotenuse is , which is or .

The Piangle is not distorted, it's just an unrolled circle. It even has the same area as its corresponding circle. Its area is 1/2*b*h = = .

Proof that I discovered this: the hypotenuse = 6.3622651. Googling that doesn't return anything about the triangle.

Edited by The Piangle

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Any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r.

He didn't give it a name though.

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Apparently this has been defined before: https://betterexplained.com/guides/calculus/

5 minutes ago, Endy0816 said:

He didn't give it a name though.

I don't know if he realized that each slice of the triangle corresponds to consecutive rings of the circle. Also we only need one triangle if we just measure it in radians

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I never knew this. Well done for working it out, even if it is two and half thousand years too late!

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I do not know how the people discourage one,s attempt ,well done keep it up DO IT IN DETAILS.....

I know few people ................( habit of Barking).....

like andy pendy.....

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Good job. Keep up the great work. I don’t think it is a thousand years old. You have just used an angle and may have unknowingly drawn an involute which is a type of logarithmic spiral. (In case you don’t know I love logarithmic spirals.) When you unroll each outline to the right; Can someone in this forum tell me if it is a linear representation of the involute? I have seen something graphical, similar to this unroll in a math reference. I can’t remember where, but I believe it was for gears.

I don’t have any pictures of a logarithmic spiral to share yet. I want to be sure it relates to your post. But even if this work is rediscovered, it doesn’t mean you can’t relate it to something new.

What I think you should try is to “put space between your Pi angles.” What I mean by this is that having a “series” between where 1/3 r and 2/3 r and r would change the shape of the involute to a special logarithmic spiral.

I know it sounds like I’m talking babble, but I am not. If you are confused on what I am trying to say, let me know and I will try and describe my idea better. Simple put I would shift the new larger angles a distance (determined by a series) across the x-axis from the original triangle that was at the origin.

This way you can craft series and describe them in a logarithmic spiral.

I will post a picture of an ellipse determined by angles. It is not a logarithmic spiral, but it will demonstrate using angles to determine geometry. I will try to work on drawing a graphic representation of the logarithmic spiral I describe here. But this will work till then.

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