A solid acid having a core of calixarene or calix resorcinarene. The solid acid is an ion conducting compound in which at least one of the hydroxyl groups is substituted by an organic group having a cation exchange group at a terminal end, a polymer electrolyte membrane including the same, and a fuel cell using the polymer electrolyte membrane. The polymer electrolyte membrane can provide low methanol crossover and high ionic conductivity. Accordingly, a fuel cell having high efficiency can be obtained by using the polymer electrolyte membrane.