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Sponges, the basalmost members of the animal kingdom, exhibit a range of complex architectures in which microfluidic channels connect multitudes of spherical chambers lined with choanocytes, flagellated filter-feeding cells. Choanocyte chambers can possess scores or even hundreds of such cells, which drive complex flows entering through porous walls and exiting into the sponge channels. One of the mysteries of the choanocyte chamber is its spherical shape, as it seems inappropriate for inducing directional transport since many choanocyte flagella beat in opposition to such a flow. Here we combine direct imaging of choanocyte chambers in living sponges with computational studies of many-flagella models to understand the connection between chamber architecture and directional flow. We find that those flagella that beat against the flow play a key role in raising the pressure inside the choanocyte chamber, with the result that the mechanical pumping efficiency, calculated from the pressure rise and flow rate, reaches a maximum at a small outlet opening angle. Comparison between experimental observations and the results of numerical simulations reveal that the chamber diameter, flagellar wave number and the outlet opening angle of the freshwater sponge $E. muelleri$, as well as several other species, are related in a manner that maximizes the mechanical pumping efficiency. These results indicate the subtle balances at play during morphogenesis of choanocyte chambers, and give insights into the physiology and body design of sponges., Comment: 13 pages, 13 figures