A circle is inscribed in an equilateral triangle. If the area of the circle is 462 cm$$^2$$, then the perimeter (in cm) of the triangle is

In an equilateral triangle, the incenter, orthocenter, median are all coincident. 
Thus, the center of the inscribed circle is also the centroid of the triangle and the radius of the circle is the smaller division of the median which gets cut by the centroid. The centroid divides the median in the ratio of 2:1. Thus, we can say that the height of the triangle will be 3 times the radius of the inscribed circle.

Radius of the circle = sqrt (462*7/22) = $$7\sqrt{\ 3}$$
Height of the triangle = $$21\sqrt{\ 3}$$

Now, Height of triangle = $$\frac{\sqrt{\ 3}}{2}\cdot Side\ of\ triangle$$
Hence, Side of triangle = 42 cm and thus, perimeter = 126 cm