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March 23rd, 2019, 08:44 AM   #1
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Question The Area of the region of the Equilateral Triangle that lies Outside the Circle

I'm considering a circle of diameter D superimposed over the equilateral triangle of side length D, with their centres coinciding.

What is the total area of the three regions of the triangle that lie outside the circle?
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March 23rd, 2019, 03:55 PM   #2
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Hopefully, I interpreted your problem statement correctly.

I looked at this problem to see if there was a way to calculate the desired area by means of elementary geometry. If there is, someone will no doubt point out a simpler method.

Having failed any elementary method, I turned to using the calculus of polar graphs to determine it.

I can tell you the exact solution is not nice as far as simplification is concerned.

If you are interested, I determined the total area of the three small "triangular" looking regions to be ...

$A = \dfrac{\pi+2(\sqrt{3}-\sqrt{2})-6\arctan\left(\frac{\sqrt{2}}{2}\right)}{8} \cdot D^2$

... attached is a sketch of the diagram used.
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