Ratio of areas of incircle and circumcircle of an equilateral triangle is?1:41:22:12:3

The ratio of the areas of the incircle and circumcircle of an equilateral triangle is 1:4.

Here's the step-by-step solution:

1. The radius (r) of the incircle of an equilateral triangle with side length (a) is given by the formula r = a/2√3.

2. The radius (R) of the circumcircle of the same equilateral triangle is given by the formula R = a/√3.

3. The area (A) of a circle with radius r is given by the formula A = πr². Therefore, the area of the incircle (Ai) is Ai = π(a/2√3)² = πa²/12 and the area of the circumcircle (Ac) is Ac = π(a/√3)² = πa²/3.

4. The ratio of the areas of the incircle and circumcircle is therefore Ai/Ac = (πa²/12) / (πa²/3) = 1/4.