To any compactly supported, area preserving, piecewise linear homeomorphism of the plane is associated a relation in $K_2$ of the smallest field whose elements are needed to write the homeomorphism.Using a formula of J. Morita, we show how to calculate the relation, in some simple cases. As applications, a “reciprocity” formula for a pair of triangles in the plane, and some explicit elements of torsion in $K_2$ of certain function fields are found.