  In this paper we consider a semitetrad covariant decomposition of spherically
symmetric spacetimes and find a governing hyperbolic equation of the Gaussian
curvature of two dimensional spherical shells, that emerges due to the
decomposition. The restoration factor of this hyperbolic travelling wave
equation allows us to construct a geometric measure of complexity. This measure
depends critically on the Gaussian curvature, and we demonstrate this geometric
connection to complexity for the first time. We illustrate the utility of this
measure by classifying well known spherically symmetric metrics with different
matter distributions. We also define an order structure on the set of all
spherically symmetric spacetimes, according to their complexity and physical
properties.
