Refer to the figure.
We are looking for the area of the sector of a circle as shown in the figure shaded with green color.Â
The area of a sector of a circle can be calculated using the formula [tex]A=\frac{1}{2}r^2sin\left(\theta \right)[/tex]
   where r=radius, and θ=central angle (in radians)
The central angle of the given sector is just one-third of a full circle (2Ï€). That isÂ
   [tex]\theta =\frac{2\pi }{3}[/tex]
Now, to solve for the radius of the circle, we can use the formula
   [tex]R=\frac{abc}{4A}[/tex]
     where R is the radius of the circumscribed circle; a,b, and c are the sides of the triangle; and A is the area of the triangle.
The area of the equilateral triangle can be solved using the formula [tex]A=\frac{\sqrt{3}}{4}a^2[/tex]. That isÂ
   [tex]A=\frac{\sqrt{3}}{4}a^2=\frac{\sqrt{3}}{4}\left(2\sqrt{3}\right)^2=3\sqrt{3}[/tex]
Now, we substitute this area in the formula to solve for the radius of the circle.Â
   [tex]R=\frac{abc}{4A}=\frac{\left(2\sqrt{3}\right)^3}{4\left(3\sqrt{3}\right)}=2[/tex]
Finally, we can solve for the area of the sector by substituting the values of the angle θ, and the radius.
   [tex]A=\frac{1}{2}r^2\theta =\frac{1}{2}\left(2\right)^2\left(\frac{2\pi }{3}\right)=\frac{4\pi }{3}\:square\:units[/tex]
