Intersection of a plane and a sphere ($\rho =\sqrt{r^2-d^2}$)

For a sphere of radius $r>0$ and a genuine plane at perpendicular distance $d\ge 0$ from its centre, the intersection is a circle of radius $\rho$ when $d<r$, a single point when $d=r$, and empty when $d>r$. The circle centre is the orthogonal projection of the sphere centre onto the plane. Decomposing a radius into its perpendicular component of length $d$ and its planar component of length $\rho$ proves the formula by the Pythagorean theorem.