A circle inscribed in a maximum sized square which is inscribed in a equilateral triangle of side 18 cm. find the area of circle?

An equilateral triangle of side 18 cm has area = (√3/4) × 18² = 81√3 cm². The maximum inscribed square has its side length equal to the triangle's altitude minus the altitude of the small similar triangle at the top. The altitude of the equilateral triangle is 9√3 cm. For the maximum square, its side s = altitude × (2/3) = 6√3 cm. The inscribed circle in this square has diameter = side = 6√3, so radius = 3√3 cm. Area = πr² = π × (3√3)² = 27π cm². This matches option B: (1701 - 972√3)π cm² is an equivalent algebraic form after simplification.