The radius inscribed in an equilateral triangle of a circle is 2 cm. Find the perimeter and radius of the circumscribed circle.

By condition, the radius of the inscribed circle in an equilateral triangle is known. From the radius formula, we find the side of the triangle:
r = a / 2√3 → a = r * 2√3 = 2 * 2√3 = 4√3 (cm).
The side of the triangle is known, we find the perimeter:
P = 3 * a = 3 * 4√3 = 12√3 (cm).
The radius of the circumscribed circle is found by the formula:
R = a / √3 = 4√3 / √3 = 4 (cm).
Answer: the perimeter is 12√3 cm, the radius of the circumscribed circle is 4 cm.