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The Piangle is a unique triangle. Every circle has its unique Piangle. To create a Piangle, you cut a circle along its bottom radius, then you unroll the left side of the circle up and over to the right until it’s all lying flat to the right of the cut. You should end up with a right triangle whose left side is equal to the circle’s radius. How large are each of a Piangle’s sides relative to each other?

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    $\begingroup$ I am so confused by this question... is this actually a geometry puzzle? Whats the bottom radius? As far as I can tell, you have a isosceles triangle with 2 sides both equal to half the circumference and 1 side equal to the radius? $\endgroup$ – Shadowzee Nov 23 '18 at 3:56
  • $\begingroup$ Imagine you have a roll of toilet paper, without a hole through the center (so all paper). Now just face the top of the roll toward you and cut from the center downward. Then grab the bottom left side, where you just cut, and throw it up and over to the right. This should leave you with a wedge/triangle shape. $\endgroup$ – Joseph Nov 23 '18 at 4:03
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    $\begingroup$ This question would greatly benefit from adding images $\endgroup$ – kanoo Nov 23 '18 at 17:23
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Supposing I understood the idea correctly, the legs are

$\text r$ and $2\pi\text r$ (the radius and the circumference of the circle)

which makes the area of the triangle

$ \frac 1 2 \times \text r \times 2\pi\text r =\pi \text r^2$, which is good, since the operation is supposed to preserve the area

And so the hypotenuse becomes

$\sqrt{\text r^2 + (2\pi\text r)^2} = \sqrt{1+4\pi^2} \times \text r$

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  • $\begingroup$ Nicely done. Correct $\endgroup$ – Joseph Nov 23 '18 at 12:07
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The original left and right sides are R*pi and the new side joining them is R, where R is the radius.

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If r is the radius of the circle, the bottom side of the triangle will be the perimeter of the circle so, its length is 2πr. Since the "left" side of the triangle is of length r, the third side (hypotenuse) will be √(r^2+(2πr)^2).

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