this term describes the size of a rifle or handgun bore. this term also describes the size of cartridges used with different bores. what is the term?

The term you’re looking for is caliber.

Caliber describes:

• The size of a rifle or handgun bore (the internal diameter of the barrel).

• The size of the cartridges designed to be used in bores of that diameter.

For example, a “.270 caliber” rifle and “.270 caliber” ammunition are matched by this shared caliber designation.