How to find the area if you don't know the lengths of a triangle?

Given that the radius is 2, so we let the three radii touches the center of the circle, let it be $O$ , and the vertices $A, B, C$ . Then, we extend $OA, OB, OC$ to meet the sides $BC, AC, BA$ at $D, E, F$ respectively.

Since it is an equilateral triangle, so we have the property that the altitude of the triangle coincides with the median and the angle bisector. Therefore, $\angle OCB = \angle OCD = \angle OBD = \angle OBF = \angle OAF = \angle OAE = 30^{\circ}$ , implying that $EC = DC = BD = BF = AF = AE =\sqrt3$ , since $\cos 30^{\circ} = \frac{\sqrt3}{2}$ . So, the sides of each side is $2\sqrt3$ . Since I am lazy to find the height, so I use the formula,

$x=\frac{a^2\sqrt3}{4}$

where $a$ is any side of the triangle, and $x$ is the area. So, substituting $a=2\sqrt3$ , we have

$x =\frac{(2\sqrt3)^2\sqrt3}{4}$

$x =3\sqrt3$

$\implies x^2 = \boxed{27}$

Q.E.D.