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.
Best Answer
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, $$\frac{4}{\sin(60^\circ)}=2R\Rightarrow R=\frac{4}{3}\sqrt 3.$$