  As a foundation of modern physics, uncertainty relations describe an ultimate
limit for the measurement uncertainty of incompatible observables.
Traditionally, uncertain relations are formulated by mathematical bounds for a
specific state. Here we present a method for geometrically characterizing
uncertainty relations as an entire area of variances of the observables,
ranging over all possible input states. We find that for the pair of position
$x$ and momentum $p$ operators, Heisenberg's uncertainty principle points
exactly to the area of the variances of $x$ and $p$. Moreover, for
finite-dimensional systems, we prove that the corresponding area is necessarily
semialgebraic; in other words, this set can be represented via finite
polynomial equations and inequalities, or any finite union of such sets. In
particular, we give the analytical characterization of the areas of variances
of (a) a pair of one-qubit observables, (b) a pair of projective observables
for arbitrary dimension, and give the first experimental observation of such
areas in a photonic system.
