We have the formula for the surface area of a cylinder:
SA = 2πrh + 2πr^2
Given that the surface area is 378π cm^2 and the radius is 7 cm, we can plug in these values and solve for the height:
378π = 2π(7)(h) + 2π(7)^2
378π = 14πh + 98π
280π = 14πh
h = 20 cm
Therefore, the height of the cylinder is 20 cm.