Three small identical circles are inscribed inside an equilateral triangle with length $$10\sqrt{3}$$ cm as shown in the figure. The radius of each small circle is 2 cm A big circle touches these three circles as shown in the figure. Find the ratio of the area of the big circle with that of the area of the small circle. (figure not as per scale)

Height of triangle = $$\left(10\sqrt{3}\right)\times\ \left(\frac{\sqrt{3}}{2}\right)=15$$

Distance between the centroid of the triangle and the vertex= $$15\times\ \frac{2}{3}=10$$

If we assume a triangle with the small circle as inradius, then the distance between the vertex to the point where the small circle and big circle touch each other becomes the height of that triangle. Since in-radius is one-third of the height of the triangle, the height can be calculated as $$2\times3=6$$

The radius of the bigger circle = Distance between centroid to vertex - the distance between the vertex to the point where the small circle and big circle touch each other  = 10 - 6 = 4cm

The ratio of areas will be the square of the ratio of radius, which is $$4^2\ :\ 2^2\ =\ 16\ :\ 4\ =\ 4\ :\ 1$$.

Hence, the correct answer is option A.