The problem of the shape of a random object such as a flexible polymer chain was first tackled by Kuhn nearly thirty years after the answer to the probability distribution of its size was publicly sought for by Pearson in 1905. Since then, significant progress in the field has been made, but the important task of evaluating both analytically and accurately averaged individual principal components of the shape or inertia tensor for a walk of a certain architectural type remains unfinished. We have recently developed a new and general formalism for both exact and approximate calculations of these and other averages such as asphericity and prolateness parameters, which is illustrated here for an end-looped random walk and a selfavoiding or Edwards chain. We find that this combined open and closed random walk has surprisingly larger shape asymmetry than a simply open walk despite its smaller size.