The operation of cutting a cylindrical hole in a sheet of metal by the punch and die is called

Punching is a sheet metal operation in which a cylindrical hole or any other desired shape is cut out from a metal sheet using a punch and die set.

In this process, the punch applies pressure on the sheet metal, forcing the material into the die opening. The portion of the metal that is removed (the slug) is considered scrap, and the remaining sheet with the hole is the useful part.

Piercing is a similar process, but it generally refers to making small holes or slots without removing significant material, often as part of a larger forming operation.

Blanking is the reverse of punching. In blanking, the cut-out portion (blank) is the useful part, while the remaining sheet is scrap.

Shearing refers to the cutting of a sheet along a straight line using shearing blades, not by punch and die action.

Therefore, the operation of cutting a cylindrical hole in a sheet of metal using a punch and die is known as punching.

Hence, the correct answer is Option C: Punching.