  Hadwiger's theorem is a variant of Helly-type theorems involving common
transversals to families of convex sets instead of common intersections. In
this paper, we obtain a quantitative version of Hadwiger's theorem on the
plane: given an ordered family of pairwise disjoint and compact convex sets in
$\mathbb{R}^2$ and any real-valued monotone function on convex subsets of
$\mathbb{R}^2,$ if every three sets have a common transversal, respecting the
order, such that the intersection of the sets with each half-plane defined by
the transversal are valued at least (or at most) some constant $\alpha,$ then
the entire family has a common transversal with the same property. Unlike
previous generalizations of Hadwiger's theorem, we prove that disjointness is
necessary for the quantitative case. We also prove colorful versions of our
results.
