How many sides does a circle have?
My son answered "infinate" on a test and was corrected. The school insists the answer is 0.

Thanks,
georganne

There are two sources of ambiguity here.

1. The word CIRCLE sometimes refers to a circular disk (because it is common to talk about "the area of a circle"), but often the word refers only to the boundary of the disk (namely, the points that are equally distant from the point at the center of the figure).
   
   
2. The meaning of the word SIDE depends on its context. A polygon in the plane has, by definition, n vertices and each pair of consecutive vertices are joined by a side. It is clear that a polygon with n vertices has n sides. In general, the word side depends on the dimension of the figure -- a side is always part of the boundary.

So what is the side of a circle? If you think of the circle as a disk then it it has an up-side and a down-side. If you think of it as a curve, then it has an inside and an outside. If you think of it as the limit of an n-sided regular polygon, then one can justify the answer that the circle has infinitely many infinitesimal sides. Our conclusion: