The question “Why does a duck stand on one leg?” is a lot easier to ask than to answer. And while it could be simply stated that a duck will stand on one leg to conserve body heat, that might be shortchanging you a bit. But let's start there anyway.
A number of avian species, including ducks and geese, often stand on one leg to thermoregulate (control their body temperature) during cold weather by keeping more heat close to the body, the same way a person might wrap their arms around themselves when cold. Much of our own heat escapes through our extremities – our arms, legs and especially head, where more than 40 percent of the body's heat can be lost on cold days. A lot of heat can also be lost through waterfowl's feet, so ducks and geese will either sit in an attempt to stay warm, saving up to 50 percent of their body heat, or they will stand on one leg. So how does that exposed leg stay warm? Counter-current exchange.