Assume that we are asked to find the surface area of a right circular cylinder given the radius. (Right indicates that it is perpendicular to the base.)
The surface area of the cylinder has three parts -- the circular bases and the side. The area of the bases is just the area of the circles so we use `A_(base)=pir^2` (Note that there are 2 bases.).
The area of the side -- imagine a soup can; peel off the label. What you have is a rectangle -- the height of the rectangle is the height of the can (cylinder.) The width of the rectangle is the circumference (perimeter) of the base. So the area of the side is `2pirh` .
The surface area of any right circular cylinder is given by:
`SA=2pir^2+2pirh` the area of the bases plus the area of the side.
If you just know the radius, you cannot give a numerical answer for the surface area -- you also need to know the height. Suppose the radius is 2 -- then the surface area is `SA=8pi+4pih` . So your answer will involve some multiple of h.
** A particular problem may give you enough information to solve for h. For example, maybe the cylinder is inscribed in a sphere or another shape. **
If you know the Volume you can solve for h:
`V=pir^2h` (The volume of a prism, and a right circular cylinder is a right circular prism, is found by V=Bh where B is the area of the base and h the height.)
Suppose we know that the radius is 2 and the volume is `V=36pi` . Then we can solve for h: `36pi=pi(2)^2h ==> h=9` and then we can find the surface area: `SA=2pi(2)^2+2pi(2)(9)=8pi+36pi=44pi`