  Given a triangulated surface, a polyhedral metric could be constructed by
gluing Euclidean triangles edge-to-edge. We carefully describe the construction
and prove that such a polyhedral metric is the only intrinsic metric on the
glued surface that preserves the lengths of the curves on the Euclidean
triangles. We also discuss the edge length coordinates of the Teichm\"uller
space of the polyhedral metrics on a marked surface.
