  We investigate the zeta-regularized determinant and its variation in the
presence of conical singularities, boundaries, and corners. For surfaces with
isolated conical singularities which may also have one or more smooth boundary
components, we demonstrate both a variational Polyakov formula as well as an
integrated Polyakov formula for the conformal variation of the Riemannian
metric with conformal factors which are smooth up to all singular points and
boundary components. We demonstrate the analogous result for curvilinear
polygonal domains in surfaces. We then specialize to finite circular sectors
and cones and via two independent methods obtain variational Polyakov formulas
for the dependence of the determinant on the opening angle. Notably, this
requires the conformal factor to be logarithmically singular at the vertex. We
further obtain explicit formulas for the determinant for finite circular
sectors and cones.
