Theory:

Everyday experience shows us that bodies in the liquid become lighter - they lose part of their weight.
For instance, lifting a rock inside the water is easier than lifting the same rock out side the water.
Why this happens?
Let's try to understand using the knowledge of pressure in liquids (hydrostatic pressure) and pressure force - a force that acts on the entire surface of the body.
 
Svars_udeni_a.png
 
Since the upper layers of the liquid are pressed against the lower ones by gravity, the deeper we are in the liquid, the higher the pressure created by the liquid. Its value is calculated according to the formula \( p= ρcf.gh \), where 
\(ρcf\)  - density of the liquid, 
\(h \) depth of location, and 
\(g\)  - acceleration of free fall.
 
 
Pressure force \( F\)  can be calculated knowing the pressure \( p\)  and surface area  \(S\) to which this pressure acts:
 
\(F= pS\)
 
Immerse the rectangular parallelepiped in the liquid.
 
Farhimeda celonis.png
 
The fluid will apply pressure to all faces of the body. The compressive forces on the opposite side faces will compensate for each other. The situation is different with the upper and lower faces. The compressive force on the lower face is directed upwards and can be calculated from the relationship
 
\(F2= p2S= ρcf.gh2S .\)
 
The compressive force on the upper face is directed downwards and is calculated by the formula:
 
\(F1= p1.S= ρcf.gh1.S\) where S is the area of ​​the upper and lower faces.
 
 
Since  h2>h1, then also F2>F1 , and an uncompensated force appears upwards.
 
 
This resulting, uncompensated compressive force is called the Archimedean force .
  
\( FA= F2- F1= ρcf.gh2S-ρcf.gh1S=ρcf.gS ( h2- h1)= ρcf.gSH= ρcf.g\) Where V is body volume..
 
If the body is not completely immersed in the liquid, then the formula  \(FA= ρcf.gV   \) we use only the volume of the part immersed in body fluid to sink.

Multiplication  \(ρcf.V\)  gives the mass of fluid expelled by the body  m . In turn, the mass multiplied by \(g \) gives the force of gravity of the displaced fluid, and we come to the formulation of Archimedes' law.
A body immersed in a liquid or gas is subjected to a vertical upward force which is numerically equal to the force of gravity of the liquid or gas expelled by the body:
\(FA= ρcf.gV\)    
  
Important!
 
The power of Archimedes also works in gases. However, the density of air is almost 1000 times lower than the density of water, and the corresponding air-generated force of Archimedes is also 1000 times lower than in water. We usually do not notice this slight lifting force. However, it is important aerodynamics of  balloon and airplanes.