what is the radius of curvature of a circle?
Radius of curvature is also called the radius and would not be an infinite number. If that's what you were asking, than the Radius would be half of the diameter (0.5*D)
If you were asking about the number of sides for a circle, than yes, it would be infinite.
"In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof."
The radius of curvature is the shortest distance between the sketch and its curvature center. the curvature center of a line is on the normal of the line but infinite. for two lines (not parallel ones), the point that both normals intersect, is the curvature center. a circle is a set of infinite lines that all of the lines normals intersect at one point and that point is center!