Find the length of a circle circumscribed about a regular triangle with sides 12√3cm.

Find the area of ​​the given triangle.

According to the problem statement, this triangle is equilateral and its side length is 12√3 cm.

Since each angle of an equilateral triangle is 60 °, the area of ​​this triangle is 12√3 * 12√3 * sin (60 °) / 2 = 12√3 * 12√3 * (√3 / 2) / 2 = 432 * √3 / 4 = 108√3 cm ^ 2.

Let R denote the radius of the circle circumscribed about this triangle.

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

108√3 = 12√3 * 12√3 * 12√3 / (4R),

solving which, we get:

108√3 = 12√3 * 12√3 * 3√3 / R;

R = 12√3 * 12√3 * 3√3 / (108√3);

R = 12√3 * 12√3 * 3/108

R = 12√3 * 12√3 / 36;

R = 432/36;

R = 12 cm.

Find the circumference:

2pR = 2p * 12 = 24p cm.

Answer: 24p cm.