We consider a semitetrad covariant decomposition of spherically symmetric spacetimes, and find a governing hyperbolic equation for the Gaussian curvature of two dimensional spherical shells, that emerges from 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.