A regular hexagon and a square are inscribed in a circle. The area of the hexagon is D. Find the side and area of the square.

The large diagonals of a regular hexagon divide it into six equal, equilateral triangles. Then Svom = D / 6 cm2.
Through the area of a regular triangle, we determine the length of its side.
Swam = ОВ2 * √3 / 4.
ОВ2 = 4 * Svom / √3 = 4 * (D / 6) / √3 = 4 * √3 * D / 6 * 3 = 2 * D * √3 / 9.
In a right-angled triangle AOB, AO = ОВ = (2 * D * √3 / 9) cm, then AB ^ 2 = ОА ^ 2 + ОВ ^ 2 = (2 * D * √3 / 9) + (2 * D * √3 / 9) = 4 * D * √3 / 9.
AB = √ (4 * D * √3 / 9) = (2/3) * √ (D * √3) cm.
Then Savsd = AB2 = ((2/3) * √ (D * √3)) 2 = 4 * √3 * D / 9 cm2.
Answer: The side of the square is (2/3) * √ (D * √3) cm, the area of the square is 4 * √3 * D / 9 cm2.