  In this paper, we characterise ovoidal cones by their intersection numbers.
We first show that a set of points of $\mathrm{PG}(4,q)$ which intersects
planes in $1$, $q+1$ or $2q+1$ points is either an ovoidal cone or a parabolic
quadric, unless $q=3$, in which case also a sporadic example with automorphism
group $M_{11}$ exists. We then show that a set of points of $\mathrm{PG}(4,q)$
which blocks all planes and intersects solids in $q+1$, $q^2+1$ or $q^2+q+1$
points is a plane or an ovoidal cone, and determine all examples that arise
when the blocking condition is omitted.
