The perimeter of a regular hexagon inscribed in a circle is 48 cm. Find the side of a square inscribed in the same circle.

Let’s draw diagonals connecting opposite sides of the hexagon.
They intersect at point O, which is the center of the circumcircle.
We get 6 symmetrical triangles, angle (AOB) = 360/6 = 60 degrees.
Because AO = BO, then angle (OAB) = angle (BOA) = 120/2 = 60 degrees.
Therefore, triangle AOB is equilateral
AB = P / 6 = 48/6 = 8 cm.
AO = 8 cm – radius of the circle.
Then the diameter of the circle is 16 cm.
Consider now the inscribed square.
Its diagonal AC is equal to the diameter of the circle, i.e. AC = 16 cm.
By the Pythagorean theorem, the side of a square is 16 / √ (2) or 8 * √ (2).
Answer: 8 √ (2).