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

Let the length of the side of the triangle be X cm, and the side of the square is Y cm.

Determine the radius of a circle circumscribed around a regular triangle.

R = ОА = X * √3 / 3 cm.

A regular quadrilateral AKMD is a square, then its diagonals are equal and intersect at right angles, then triangle AOD is rectangular, and OD = OA = X * √3 / 3 cm.

Then, by the Pythagorean theorem, AD ^ 2 = OA ^ 2 + OD ^ 2 = (X * √3 / 3) 2 + (X * √3 / 3) 2 = X ^ 2/3 + X ^ 2/3 = 2 * X ^ 2/3.

AD = Y = X * √2 / √3 cm.

Then X / Y = √3 / √2.

Answer: The aspect ratio is √3 / √2.