Your word "define" is not the word I would use. Most people would DEFINE a circle in terms of a point (called the center) and a length (called the radius) as the set of all points in the plane whose distance from the center equals the radius. We then would prove as a theorem that the circumference of the circle is equal in length to the limit of the perimeters of regular n-sided polygons inscribed in (or circumscribed about) the circle as n goes to infinity. Similarly, the area enclosed by the circle equals the limit of the areas of the inscribed regular polygons.
You are right that the same sort of relationship exists between cylinders and prisms.