The length of a circle circumscribed about a regular triangle is 12пcm. Find the perimeter of the triangle.

The formula for the circumference of a circle: L = 2ПR, where R is the radius of the circle. L = 12П So, 12П = 2ПR => R = 6 cm Let’s call the triangle ABC Since the triangle is regular, the median, bisector and height are one and the same. And yet, all the medians of the triangle intersect at one point, which is called the center of the triangle O (respectively, the circle, because OA = OB = OS) and is divided by this point into two parts in a ratio of 2: 1 counting from the vertex. This means that 6 cm is 2 parts. Then 1 part = 3 cm => that the entire median of the triangle is 9 cm Median AH (let’s call it that) is perpendicular to BC. It turns out a right-angled triangle ONV in which the hypotenuse (6 cm) and one leg (3 cm) are known. By the Pythagorean theorem, we find the third leg HB: HB = √ (6 ^ 2 – 3 ^ 2) = √ (36-9) = √27 = 3√3 (cm) Then BC = 2 * 3√3 = 6√3 ( cm) That’s all, we found one side of the triangle, and since ABC is correct, its P = 3 * 6√3 = 18√3 cm Answer: 18√3 cm