We present a method for the classification of 2-D partial shapes using Fourier descriptors. We formulate the problem as one of estimating the Fourier descriptors of the unknown complete shape from the observations derived from an arbitrarily rotated and scaled shape with missing segments. The method used for obtaining the estimates of the Fourier descriptors minimizes a sum of two terms; the first term of which is a least square fit to the given data subject to the condition that the number of missing boundary points is not known and the second term is the perimeter2/area of the unknown shape. Experiments with synthetic and real boundaries show that estimates closer to the true values of Fourier descriptors of complete boundaries are obtained. Also, classification experiments performed using real boundaries indicate that reasonable classification accuracies are obtained even when 20-30 percent of the data is missing.