# Find the total surface area of a regular hexagonal prism if its base area is 54 * √ (3) cm², and the volume is 324 cm³.

Knowing the volume of the prism and the area of its base, we determine the height of the prism.

V = Sosn * AA1.

АА1 = V / Sсн = 324/54 * √3 = 6 / √3 = 2 * √3 cm.

The area of a regular hexagon is: Sbn = 3 * √3 * a^2 / 2, where a is the side of the hexagon.

a^2 = 2 * Sb / 3 * √3 = 2 * 54 * √3 / 3 * √3 = 36.

a = AB = 6 cm.

The side faces of a regular hexagonal pyramid are equal triangles, then S side = 6 * AB * AA1 = 6 * 6 * 2 * √3 = 72 * √3 cm2.

Then Sпов = 2 * Sсн + S side = 2 * 54 * √3 + 72 * √3 = 180 * √3 cm2.

Answer: The total surface area of the prism is 180 * √3 cm2.

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