I don't know Macedonian, but my guess is that the same thing is in Russian. "Pol-" can be used only as a prefix with other nouns (and not only nouns), whereas "polovina" is an independent noun.
Am I right?
Пол[а] is used to denote half a quantity of something or a quantity and a half of something in expressions like часот е девет и пол ('it's half past one'), пола[а] кило[грам] сол ('half a kilo of salt'). Половина can technically be used in this sense (пола/половина саат/час), but the contracted form is much more common. Половина is a noun; првата половина од 2013 година ('the first half of 2013'). It can also refer to the middle back (ме боли половината) and, figuratively, the waist or figure: таа има тенка половина ('she has a slim waist'), таа има убава половина ('she has a nice figure').
In compounds, полу- is a prefix used much like the English 'semi-' (полукруг, 'semicircle'), 'demi-' (полубог, 'demigod') 'half-' (полубрат, 'half-brother') and probably others as well.