Judith is playing with triangles and circles

Judith inscribed a circle in a equilateral triangle of sides $x$ $cm$ . Then she inscribed another triangle in that circle. If Judith could do this process infinitely, the sum of the areas of all the triangles and circles would be expressed as $\dfrac{x^{2}}{k}(\dfrac{\pi}{k}+\sqrt{k})$ . Find $k$ .