The radius of the circle is 2√3 cm. Find the side of a regular triangle circumscribed around this circle.

Let us denote by a the length of the side of an equilateral triangle circumscribed around this circle.

Since each angle of an equilateral triangle is 60 °, then applying the formula for the area of ​​a triangle along two sides and the angle between them, we find the area S of this triangle:

a * a * sin (60 °) / 2 = a ^ 2 * (√3 / 2) / 2 = a ^ 2 * √3 / 4.

According to the condition of the problem, the radius of the circle is 2√3 cm.

Using the formula for the area of ​​a triangle in terms of the radius of the inscribed circle, we can compose the following equation:

(3a / 2) * 2√3 = a ^ 2 * √3 / 4,

solving which, we get:

3√3a = a ^ 2 * √3 / 4;

3√3 = a * √3 / 4;

a = 3√3 / (√3 / 4) = 3√3 * 4 / √3 = 12 cm.

Answer: 12 cm.