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The formula to calculate the surface area of a cylinder is given by:
2Ï€r(r + h), where r is the radius and h is the height.
Plugging in the values:
2Ï€(6mm)(6mm + 9mm) = 2Ï€(6mm)(15mm) = 180Ï€ mm^2 = 565.2 mm^2.
Therefore, the correct answer is 565.2mm^2.