Under the assumptions of an incompressible fluid with negligible viscosity, Bernoulli's principle states that the hydraulic energy is constant 
at any two points in the flowing liquid. Here  is fluid speed,  is the acceleration due to gravity,  is the height above some reference point,  is the pressure, and  is the density.
In order to derive Torricellis formula the first point with no index is taken at the liquids surface, and the second just outside the opening. Since the liquid is assumed to be incompressible,  is equal to  and ; both can be represented by one symbol . The pressure  and  are typically both atmospheric pressure, so . Furthermore 
 is equal to the height  of the liquid's surface over the opening:  
The velocity of the surface  can by related to the outflow velocity  by the  continuity equation , where  is the orifices cross section and  is the (cylindrical) vessels cross section. Renaming  to  (A like Aperture) gives:
Torricelli's law is obtained as a special case when the opening  is very small relative to the horizontal cross-section of the container :
Torricelli's law can only be applied when viscous effects can be neglected which is the case for water flowing out through orifices in vessels.