# The side of a regular hexagon inscribed in a circle is 81. A square is inscribed

The side of a regular hexagon inscribed in a circle is 81. A square is inscribed in the same circle. What is the area of the circle inscribed in this square?

Since a regular hexagon is inscribed in the circle, the radius of the inscribed circle is equal to the length of the side of the hexagon.

ОВ = ОА = AB = R = 81 cm.

The radius R is half the length of the diagonal of a square inscribed in a circle.

ОВ = OC = R = 81 cm.

The OBC triangle is rectangular and isosceles, then BC ^ 2 = 2 * OB ^ 2 = 2 * 81 ^ 2.

BC = 81 * √2 cm.

Sows = ОВ * ОВ / 2 = 6561/2 cm2.

Also Sovs = BC * OH / 2.

ОН = r = 2 * Sоvs / ВС = 2 * (6561/2) / 81 * √2 = 81 / √2 cm.

Then Scr = π * r ^ 2 = π * 6561/2 = 3280.5 * π cm2.

Answer: The area of the circle is 3280.5 * π cm2. One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.