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= $$15\times\ \frac{2}{3}=10$$

Since in-radius is one-third of the height of the triangle

therefore, length of the line from vertex to the point where small circle and big circle touch each other = $$2\times3=6$$

Radius of the bigger circle = 10 - 6 = 4cm

Ratio of areas will be square of the ration of radius which is 4:1.