Find the length of a circle if the area of a regular hexagon inscribed in it is 72√3 cm3.

1. By the condition of the problem, it is known that the area S of the inscribed regular hexagon is 72√3 cm².

2. Connect all the vertices of the hexagon with the center of the circle: we got 6 equilateral triangles, the side length of which is equal to the radius R of the circumscribed circle.

So the area of ​​each triangle is s = 1/6 S = 72√3: 6 = 12√3 cm², and according to the formula

s = 1/2 base to height = 1/2 * R * h.

The value of h is calculated as a leg of a right-angled triangle with an angle of 30 °:

h: R / 2 = ctg 30 °, whence h = R / 2 * ctg30 ° = R / 2 * √3.

Let’s equate two values ​​of s:

12√3 = R / 2 * R / 2 * √3, that is, 12√3 = R² / 4 * √3, whence R² / 4 = 12, which means

R = √12 * 4 = √48 = 4√3 cm.

3. Circumference L = 2 * P * 4 * √3 = 25 √3 cm.

Answer: L = 25 √3 cm.