The difference between the area of the circumscribed circle and the area of the inscribed circle of an equilateral triangle is 2156 sq. cm. What is the area of the equilateral triangle?

Let radius of incircle = $$r$$, => Radius of circumcircle = $$2r$$

Difference in area = $$\pi [(2r)^2 - (r)^2] = 2156$$

=> $$3 \times \frac{22}{7} \times r^2 = 2156$$

=> $$r^2 = \frac{2156 \times 7}{66}$$

=> $$r = \sqrt{\frac{686}{3}}$$

Now, height of equilateral triangle = $$3 r = \frac{\sqrt{3}}{2} a$$    (where $$a$$ is side of triangle)

=> $$3 \times \sqrt{\frac{686}{3}} = \frac{\sqrt{3}}{2} a$$

=> $$a = 2 \sqrt{686}$$

$$\therefore$$ Area of triangle = $$\frac{\sqrt{3}}{4} a^2$$

= $$\frac{\sqrt{3}}{4} \times 4 \times 686 = 686 \sqrt{3} cm^2$$