The lateral profile of the energy deposition in a prototype of a lead and scintillating fiber calorimeter, constructed in the framework of the LAA project at CERN, has been measured for both electromagnetic and hadronic showers in the energy range from 5 to 150 GeV. The distributions are well described by analytic functions whose parameters allow one to determine the radial scaling of the shower development. In the electromagnetic case, the data are compared to Monte Carlo calculations. Estimates of the lateral leakage outside of the detector are made as well as calculations of the average π0 content of hadronic showers and event-to-event fluctuations in this electromagnetic component. A method is developed whereby knowledge of the hadronic shower profile can be used to determine the relative calibration constants of neighboring towers in such a calorimeter.