The radius of a circle inscribed in a regular hexagon is 8√3 cm. Find the diameter of a circle inscribed around this hexagon?

Since a regular hexagon is inscribed in a circle, its sides rest on arcs, the degree measures of which are 360/60 = 60.

Then the central angle AOB = 60.

In the triangle AOB OA = OB = R = 8 * √3 cm, and one of the acute angles is 60, then the triangle AOB is equilateral and AB = OA = OB = 8 * √3 cm.

The height OH of the equilateral triangle AOB is also its median, then BH = AH = AB / 2 = 8 * √3 / 2 = 4 * √3 cm.

In the right-angled triangle VON, according to the Pythagorean theorem, we determine the length of the hypotenuse OB, which is the radius of the circumscribed circle.

ОВ ^ 2 = ОН ^ 2 + ВН ^ 2 = 192 + 48 = 240.

ОВ = 4 * √15 cm.

Then D = ВС = 2 * ОВ = 8 * √15 cm.

Answer: The diameter of the circumscribed circle is 8 * √15 cm.