  We show that any immersion, which is not a covering of an embedded
2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable
hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally
geodesic boundary, called the "turnover core,'' whose volume is bounded from
above by a function depending only on the area of the given turnover.
Furthermore, we show that, for a given type of turnover, there are only
finitely many possibilities for the turnover core. As a corollary, if the
volume of a complete orientable hyperbolic 3-orbifold is at least 2\pi and if
the fundamental group of the orbifold contains the fundamental group of a
hyperbolic turnover (i.e., a triangle group), then the orbifold contains an
embedded hyperbolic turnover.
