Fan doesnt kick on and it sttarts to overheat?

I have a 92 firebird 305tbi. The fan doesnt kick on it get to about 240 degrees than I just shut the car off. This only happens when it idles. Otherwise the gauges stay where the are suppose to they dont go up or down after the car is at temp its suppose to be at. The fan kicks on when I turn the a/c on. I dont know if it the fan, temperature sensor or relay. I would sure like some input on this so I can figure it out. Also on one website they say that they run hot. Do I dare let it get above 240 degrees to see if the fan kicks on? It overheated a couple of months ago at idle and had to replace radiator and thermostat. The fan use to kick on when I turned the key on when I first bought the car right before overheating and now nothing.