1) Surface Area of a cylinder is equal to the two circular ends (top and bottom) as well as the side. If you think of a cylinder as a roll of paper towels, unwrapping the first layer yields a section of paper towels which is a rectangle, as tall as the roll (H) and long enough to wrap all the way around the circumference of the cylinder (L = C = 2 pi r). So,
S = 2 (pi r2) + 2 pi r H = 2 pi r (r+H)
= 2 pi (4) (4+8) = 96 pi
2) Surface area of a rectangular prism (cube like box) is equal to all 6 faces, front and back, left and right, top and bottom. Due to its symmetry, each pair of opposite faces is the exact same size. Thus,
S = 2 (L x W) + 2 (W x H) + 2 (H x L) = 2 (LW + WH + HL)
= 2 ( 8x12 + 12x14 + 14x8 ) = 2 (376) = 752
3) Combining the figures while removing the circular ends of the cylinder, to create a hollow storage tank of some sort, requires removing both circular ends from the cylinder, as well as an equal circular section to create the opening into the rectangular prism. Therefore,
S = 2 pi r (r+H) + 2 (LxW + WxH + HxL) - 3 (pi r2) = 2 pi r (H - 1/2 r) + 2(LW + WH + HL)
= 2 pi 4 (8 - 4/2) + 752 = 752 + 48 pi
If the cylinder is simply welded atop the closed box without cutting an opening down into it, or if the top is kept on the cylinder to create a closed volume, then
S = 2 pi r H + box = 752 + 64 pi
