The calculation of the resistance of a thin film rectangle depends on the location and size of the contact pads in a complex way. Two patterns of interest are solved in terms of two functions, each of which is a function of only two variables. These functions are given in tabular, graphical and equation form. The resulting elliptic functions are expanded in infinite series, and useful approximations are given in limiting cases. These same functions are then used to calculate the resistances of twenty-one other patterns of interest, including some rectangles with three and four contacts.