Please explain whether or not a cylinder has to have a circular base.
My answer is yes because if the top portion is circular, then the base needs to be circular in order to be parallel or have a lateral surface.
Is my answer even close to being correct?
It just says that the simple closed curves along with their interiors are the bases of the cylinder and the remaining points constitute the lateral surface of the cylinder. If a base of a cylinder is a circular region, the cylinder is a circular cylinder.
This explanation is above my head, so I was trying to figure out a way to simplify the answer because this seems to be a little confusing.
The Wikipedia entry for cylinder suggests that the use of the unqualified term cylinder should mean an ordinary right circular cylinder but that the word is used more broadly in differential geometry. Collins dictionary of mathematics has two geometric definitions, the first is "solid bounded by two parallel planes that moves around a fixed closed curve at a fixed angle to the planes". Collins also suggests that unqualified "cylinder" usually means right circular cylinder. The second definition is that it is a surface formed by a line segment moving round a closed plane curve at a fixed angle to its plane.
My old OED has a pseudo-mathematical definition which is probably intended to mean the same as the first Collins definition.
Euclid's definition of a cylinder amounts to saying that it is the solid you get by fixing one side of a rectangle and describing a circle with one of the non-fixed vertices, i.e. a right circular cylinder. (Similarly his definition of cone is the solid you get by fixing one of the legs of a right-angled triangle and describing a circle with the other vertex).
I'd hope that any paper or book that was going to use cylinder to means something other than a right circular cylinder would say so very clearly near the front.