= Intersection of a plane and a sphere
{title2=$\rho=\sqrt{r^2-d^2}$}
For a <sphere> of radius $r>0$ and a genuine <plane> at perpendicular <distance> $d\ge0$ 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>.
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