It depends on what you mean by "run out" but there is probably a leak in the tank.
Sounds like your water level is running low. Check the level in the radiator reservoir You may also have a leak.
Where is it leaking from? Could leak from the radiator, hoses, or water pump.
possibly a leak in cooling system
blown head gasket
No. The only times the radiator would leak is if it was damaged, defective, or corroded.
Find leak, repair/replace cause.
Could be a hose leak, a radiator leak or a head gakket leak. The water in the reservoir will get "sucked" into the cooling system if the level get low. Look for water on the ground or signs of spray under the hood. If it's a head gasket, look for water dripping from the exhaust pipe on start-up and/or steam. That would indicate that water is getting into the combustion chamber.
A radiator cap would not cause a leak! However, if the radiator cap is holding pressure, other weak parts could signal issues in need of attention.
It probably has a leak. Here's something to check - the radiator has a tube coming out to the reservoir, so when it gets hot and expands the coolant goes into the reservoir and should go back into the radiator when it cools, but if your reservoir has a crack and the coolant leaks out, well it's gone and can't go back into the radiator - this would cause a lot of loss of coolant.
It sounds like there's a leak somewhere. If it's not in the cap, it could be in the hose, or the connection, or the radiator itself. If you can't figure this out on your own, take it to a mechanic.
You have a leak in the cooling system. Could be water pump, hoses, radiator, headgasket, etc.