Regular quadrangle and hexagon are inscribed in the circle. What is the ratio of the sides of a quadrilateral to a hexagon?

Let the length of the side of the quadrangle be X cm, and the length of the side of the hexagon equal to Y cm.

A regular quadrilateral is a square, then its diagonal is equal to the diameter of the circle circumscribed around it. From a right-angled triangle ABC, AC ^ 2 = D ^ 2 = X ^ 2 + X ^ 2 = 2 * X ^ 2.

AC = X * √2 cm.Then OA = OB = R = D / 2 = X * √2 / 2 cm.

A regular hexagon divides the circle into six arcs, the degree of which is 360/6 = 60.

Let’s draw the radius of OM. The central angle BОМ = 60, and since ОВ = ОМ, the triangle BОМ is equilateral, and therefore BM = Y = X * √2 / 2.

Then X / Y = 2 / √2 = √2.

Answer: The aspect ratio is √2.