Semiconducting nanowires, unlike bulk, can be grown in both wurtzite and zincblende crystal phases. This unique feature allows for growth and investigation of technologically important and previously unexplored materials, such as wurtzite AlGaAs. Here we grow a series of wurtzite AlGaAs nanowires with Al content varying from 0.1 to 0.6, on silicon substrates and through a comparative structural and optical analysis we experimentally derive, for the first time, the formula for the bandgap of wurtzite AlGaAs. Moreover, bright emission and short lifetime of our nanowires suggest that wurtzite AlGaAs is a direct bandgap material.