Another proof using a parametrization:
The unit circle $S^1 = \\{(x,y) \in \mathbb{R}^2 \mid x^2+y^2=1\\}$ can be parametrized by $$\gamma: [0,2\pi) \to S^1,\ \gamma(t):=(\cos(t),\sin(t))$$ Note that $\gamma$ is a bijection (check!) and hence $S^1$ has the same cardinality as the interval $[0,2\pi)$ which again has the same cardinality as $\mathbb{R}$. Thus, the unit circle, as well as any other circle (you just need to modify the parametrization a bit), even contains uncountably many points.
There are more elementary proofs for this problem, but note that this proof via parametrizations can be used similarly for many geometric objects.