  We prove that any compact surface with constant positive curvature and
conical singularities can be decomposed into irreducible components of standard
shape, glued along geodesic arcs connecting conical singularities. This is a
spherical analog of the geometric triangulations for flat surfaces with conical
singularities. The irreducible components include not only spherical triangles
but also other interesting spherical polygons. In particular, we present the
class of \textit{half-spherical concave polygons} that are spherical polygons
without diagonals and that can be arbitrarily complicated. Finally, we
introduce the notion of core as a geometric invariant in the settings of
spherical surfaces. We use it to prove a reducibily result for spherical
surfaces with a total conical angle at least $(10g-10+5n)2\pi$.
