While calculating hoop's stress we consider projected area. Infact in bearing stress from pins in a truss something similar is applicable.Can you explain why we are taking something like this instead of the curved area?
We cut the vessel in half as shown below.
It is the vertical projection considered because the pressure inside the cylindrical vessel due to fluid acts perpendicular to the surface of the cylindrical vessel, such that differential force is perpendicular at any point. Assuming that we take the horizontal component and vertical component of each of the differential forces, all horizontal component of the differential force is actually countered on the other side of the vessel, therefore giving an equilibrium in horizontal direction.
The hoop now only considers vertical component of the force, therefore, whatever configuration of the vessel, we need to take only the vertical component, thus we only need the area of the vessel perpendicular to the force being considered.
As pointed in a comment, this analysis is similar to bearing pins in truss member in a way because in analysis of pins, we take the diameter of the pin as equal to the diameter of the hole, thus the pressure in the pin is EQUAL in the half of the pin, as the other side is unloaded. The bearing area is then calculated as the projected area because of what is explained above. In reality, this is not what happens because we tend to increase the hole diameter for ease in installation. Pressure vessels on the other hand is analyzed similar to this because fluid will tend to pressurized all directions with the same value. Thus, bearing area projection is similar in these two cases.