Aquaporin channels are able to selectively conduct water across cell membranes, with remarkable efficiency. Although molecular details are crucial to the pore performance, permeability is also strongly limited by viscous dissipation at the entrances. Could the hourglass shape of aquaporins optimize such entrance effects? We show that conical entrances with suitable opening angle can indeed provide a large increase of the channel permeability. Strikingly, the optimal opening angles compare well with the angles measured in a large variety of aquaporins, suggesting that their hourglass shape could be the result of a natural selection process toward optimal permeability. This work also provides guidelines to optimize the performances of artificial nanopores, with applications in desalination, ultrafiltration, or energy conversion.