How can two circles with different circumferences be matched point for point?

Two circles of different circumferences can be matched point for point because both sets of points are infinite and possess the same cardinality. In set theory, two infinite sets are considered to have the same size if they can be placed into a one-to-one correspondence, meaning every point on the first circle can be uniquely paired with exactly one point on the second circle.1

This concept is a fundamental aspect of transfinite numbers, which describe the sizes of infinite sets. While it may seem counterintuitive that a larger circle contains the same number of points as a smaller one, Georg Cantor demonstrated that infinite sets do not behave like finite sets; specifically, a set can have the same cardinality as one of its proper subsets.1

In the context of geometry, this mapping can be visualized by placing the two circles concentrically and drawing rays from their common center. Each ray intersects both circles at exactly one point, establishing a perfect one-to-one correspondence between the points of the two circles, regardless of their different circumferences.