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Ok so here is what I know, the circumcircle of an equilateral triangle with an area of $4\sqrt{3}$ is drawn, calculate the radius lenght of the circumcircle.

I also know that to find the radius I have to use the following formula in a triangle $abc$ use $\dfrac{1}{2}ab\sin A$ however I can't figure out how to use the area to find the lenght of ab or any side of the circle.

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Since the area of an equilateral triangle with the side length $a$ is $\frac{\sqrt 3}{4}a^2,$ we have $$4\sqrt 3=\frac{\sqrt 3}{4}a^2\Rightarrow a=4.$$

Then, letting $R$ be the radius of the circumcircle, we have, by the law of sines, $$\frac{4}{\sin(60^\circ)}=2R\Rightarrow R=\frac{4}{3}\sqrt 3.$$

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