How can two circles with different circumferences have the same number of points?

Two circles with different circumferences can contain the same number of points because they can be placed in a one-to-one correspondence, a concept fundamental to understanding infinite sets.1

If you draw two concentric circles, one with a larger radius than the other, you can pair every point $P$ on the outer circle with a unique point $P^{\prime }$ on the inner circle. This is achieved by drawing a straight line from their common center $O$ through point $P$ to its intersection with the inner circle at $P^{\prime }$. Because every point on the outer circle maps to exactly one point on the inner circle, the two sets have the same cardinality, or size.1

While intuition suggests the larger circle should contain more points, this example illustrates that for infinite sets, the whole can be equal to a proper subset of itself. This counterintuitive property led mathematicians like Georg Cantor to develop a rigorous theory of set theory to classify different sizes of infinity, known as transfinite cardinals.1