A circle is inscribed in a regular hexagon with a side of 10 cm. Find the radius, the side of the square inscribed in this circle.

Since the hexagon is regular, all its sides are equal, then they divide the circle into six equal arcs, the degree measure of each of which is equal to: arc AB = 360/6 = 60.

Let’s draw the radii ОА and ОВ of the circle. In triangle AOB, the central angle AOB is equal to the degree measure of arc AB. Angle AOB = 600, and ОА = ОВ = R. Then triangle AOB is equilateral, ОА = ОВ = AB = R = 10 cm.

The diagonal of the inscribed square is the diameter of the circle. EH = D = 2 * R = 20 cm.

In a right-angled triangle, EPН EP = PH as the sides of the square, then the acute angles of the triangle are 45.

PH = EH * Cos45 = 20 * √2 / 2 = 10 * √2 cm.

Answer: The radius of the circle is 10 cm, the side of the square is 10 * √2 cm.