  We calculate the universal part of the free energy of certain finite two-
dimensional regions at criticality by use of conformal field theory. Two
geometries are considered: a section of a circle ("pie slice") of angle \phi
and a helical staircase of finite angular (and radial) extent. We derive some
consequences for certain matrix elements of the transfer matrix and corner
transfer matrix. We examine the total free energy, including non- universal
edge free energy terms, in both cases. A new, general, Casimir instability
toward sharp corners on the boundary is found; other new instability behavior
is investigated. We show that at constant area and edge length, the rectangle
is unstable against small curvature.
