In math, most everything is a matter of definitions, and definitions are generally only valid insofar as they are useful.
In talking in a strict, mathematical sense about the nature of solid shapes, it's useful to describe three special places on the surface of the solid:
1) Sides. A side is made up of points between which there's a smooth continuity. Sometimes planar, sometimes not, but in any case there's no sharp discontinuity between points on the same side. The shape of a side is a plane area, or a nonplane surface. You can basically (if not very rigorously) say it's a region of the surface over which the first derivative of the surface exists in every direction.
2) Edges. An edge is made at the intersection of two sides, usually, and in this case it's the set of contiguous points which are all members of both side 1 and side 2. Sometimes an edge is linear, sometimes not, but in any case the edge is the discontinuity at the boundary of a side. The shape of an edge is either a line, or a nonlinear curve. You could say it's a region of the surface where the first derivative exists, but only in the direction along the edge.
3) Corners. A corner is different than an edge -- its' a set of points made up of exactly one point. Often it's the intersection of multiple edges, but it doesn't have to be -- take the tip of a cone. This is a region of the surface where the first derivative of the surface curvature doesn't exist, along any direction.
So you've got places where the surface doesn't bend sharply, places where it bends sharply in some directions but is smooth in others, and places where it bends sharply in every direction. It's relevant to distinguish between them. and in learning math, it's useful to learn to work with definitions. Using the definitions I've given above, a cylinder has no corners, but it has two edges.
It's also useful to be a creative thinker, and the kid who gets the most out of math class recognizes that the definitions are only definitions, useful as a framework for classifying things, and not gospel from the Math Doctor.
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