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.
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