Why does $ x^2+y^2=r^2 $ have uncountably many real solutions?

Question

What is exactly the reason the equation of a cirle of radius $ r $ and centered at the origin has uncountably many solutions in $\mathbb { R} $?

Answer 1

For every $x\in[-r,r]$ there is a solution $(x,\sqrt{r^2-x^2})$.

The interval $[-r,r]$ is bijective with $[0,1]$ by $t\to t/2r+1/2$.

And $[0,1]$ is uncountable by Cantor's.

Answer 2

The mapping $f\colon\mathbb{R}\to\mathbb{R}^2$ defined by $$ f(t)=\left(r\frac{1-t^2}{1+t^2},r\frac{2t}{1+t^2}\right) $$ is injective and its image is the circle with center at the origin and radius $r$, except for the point $(-r,0)$.