  We study the rigidity of polyhedral surfaces using variational principle. The
action functionals are derived from the cosine laws. The main focus of this
paper is on the cosine law for a non-triangular region bounded by three
possibly disjoint geodesics. Several of these cosine laws were first discovered
and used by Fenchel and Nielsen. By studying the derivative of the cosine laws,
we discover a uniform approach on several variational principles on polyhedral
surfaces with or without boundary. As a consequence, the work of Penner,
Bobenko-Springborn and Thurston on rigidity of polyhedral surfaces and circle
patterns are extended to a very general context.
