A regular quadrilateral is inscribed in a circle, and a regular quadrilateral is described around this circle

A regular quadrilateral is inscribed in a circle, and a regular quadrilateral is described around this circle. find the ratio of the perimeters and areas of these quadrangles.

Let the radius of the circle be R cm.

The length of the side of the described square around the circumference is: AB = 2 * R cm.

Then Р1 = 4 * 2 * R = 8 * R see.

S1 = 4 * R ^ 2 cm2.

The side of the inscribed square is determined from the BOS right-angled triangle.

ВС ^ 2 = 2 * R2.

BC = R * √2 cm.

Then the perimeter of the inscribed square is: Р ^ 2 = 4 * R * √2, the area is equal to:

S ^ 2 = 2 * R ^ 2 cm2.

Then:

Р1 / Р2 = 8 * R / 4 * R * √2 = 2 / √2 = √2.

S1 / S2 = 4 * R ^ 2/2 * R ^ 2 = 2.

Answer: The ratio of the perimeters is √2, the ratio of the areas is 2.