Surface area of prisms and cylinders:
In both cases the surface area is the lateral area plus the sum of the areas of the bases.
In the case of a prism ( usually defined as 2 n-gons in parallel planes connected by n parallelograms) you find the area of the bases and add the areas of the parallelograms that make up the "sides".
Thus if the bases are n-gons separated by segments of length h, the surface area is `SA=2(Area_"base")+sum_(i=1)^ns_i*h` where the second term is the sum of the areas of the parallelograms (each has an area of the corresponding side length times the lateral edge length)
If the bases are regular, you get `SA=2(1/2ap)+ph=ap+ph` where a is the apothem of the base, p the perimeter of the base, and h the lateral edge length.
For a cylinder, you still have twice the area of a base plus the lateral area. The bases are circles so you have `2pir^2` where r is the radius of the base.
For the lateral area, peel the paper cover off of a can -- the paper, once unfolded, is a rectangle of height h and width equal to the circumference of the can, so the lateral area is `2pirh`
So the surface area of a right cylinder is `2pir^2+2pirh`