Find the area of a regular triangle if the radius of the inscribed circle is 4 cm.

1. Find the radius of the circumscribed circle, it is twice as large as the circle inscribed in a regular triangle:

R = 2 * 4 = 8 (cm).

2. Determine the area of a regular triangle by the formula S = (3 * √ (3 * R ^ 2)) / 4, where S is the area of a regular triangle, R is the radius of the circumscribed circle, if the radius of the circumscribed circle is 8 centimeters:

S = (3 * √ (3 * 8 ^ 2)) / 4 = (3 * √ (3 * 64)) / 4 = (3 * 8√3) / 4 = (3 * 2 * 4 * √3) / 4 = 3 * 2 * √3 = 6√3 (cm ^ 2).

Answer: the area of a regular triangle is 6√3 square centimeters.