The curvature of a curve is an intrinsic property that measures the variation of the unit tangent direction. The more closed is the curve, the greater the curvature. The curvature of a line is zero.
The radius of curvature is the inverse of the curvature. It is the radius of a circle tangent to the curve that best fits at each point. Obviously changes at every point.

Verify that the given formula is correct for curves in the plane.