  We investigate the energetics of droplets sourced by the thermal fluctuations
in a system undergoing a first-order transition. In particular, we confine our
studies to two dimensions with explicit calulations in the plane and on the
sphere. Using an isoperimetric inequality from the differential geometry
literature and a theorem on the inequality's saturation, we show how geometry
informs the critical droplet size and shape. This inequality establishes a
"mean field" result for nucleated droplets. We then study the effects of
fluctuations on the interfaces of droplets in two dimensions, treating the
droplet interface as a fluctuating line. We emphasize that care is needed in
deriving the line curvature energy from the Landau-Ginzburg energy functional
and in interpreting the scalings of the nucleation rate with the size of the
droplet. We end with a comparison of nucleation in the plane and on a sphere.
