If a regular hexagon with a side of 2√3cm is described near a circle, find the side of a square inscribed in this circle

Let’s construct the segments OS and OD. The triangle СОD is equilateral, OC = ОD = СD = 2 * √3 cm.

The height OB of an equilateral triangle COD is the radius of the inscribed circle.

Then R = ОВ = СD * √3 / 2 = 2 * √3 * √3 / 2 = 3 cm.

The diameter BE of a circle is the diameter of a square inscribed in it. BE = 2 * R = 2 * 3 = 6 cm.

Triangle BAE is rectangular and isosceles, AB = AE as sides of a square.

Then BE ^ 2 = 2 * AB ^ 2.

AB ^ 2 = BE2 / 2 = 36/2 = 18.

AB = √18 = 3 * √2 cm.

Answer: The side of the inscribed square is 3 * √2 cm.