The perimeter of a regular triangle is 42√3 cm. Find the circumference inscribed in it.

Since the triangle ABC is equilateral, and its perimeter is 42 * √3 cm, the length of the sides of the triangle is: AB = BC = AC = P / 2 = 42 * √3 / 3 = 14 * √3 cm.

Let’s build the height BH, which is also its median, then AH = AC / 2 = 14 * √3 / 2 = 7 * √3 cm.

In a right-angled triangle ABH, according to the Pythagorean theorem, we determine the length of the leg BH.

BH ^ 2 = AB ^ 2 – AH ^ 2 = 588 – 147 = 441.

BH = 21 cm.

Point O is the point of intersection of the medians, which are divided in it in a ratio of 2/1.

OH is the radius of the inscribed circle, then OH = BH / 3 = 21/3 = 7 cm.

Determine the length of the inscribed circle. C = π * 2 * OH = 14 * π cm.

Answer: The circumference is 14 * π cm.