General aviation Flight Instructor pilot a near miss with a helicopter in the traffic pattern at a non-towered airport. The instructor took evasive action to avoid a collision.

Narrative

My student and I were on downwind for runway XX at ZZZ. There was traffic on final so we extended our downwind for spacing. My student was flying and I saw a helicopter about a mile away heading north west at about the same altitude; at the time no factor based on their flight path. Just as we were getting ready to turn base we saw the helicopter turn right towards us on a path to cross right over our flight path now heading east. The helicopter was more to our right so I took the flight controls from my student and turned to the left to avoid the helicopter traffic. Once out of the way I was looking to turn onto final approach when the helicopter continued behind us now intersecting with the final approach path for runway XX. Once I was clear to turn in for final; I realized I had skimmed the ZZZ1 class C surface area while avoiding the traffic. The helicopter ended up flying on final before crossing right over the approach end of runway XX and heading to the southeast. We did not hear one radio call from the helicopter.I was told that there is a hospital nearby and this might've been one of their med evac flights. My suggestion would be to make contact with them and ask if they could check in on frequency when transitioning the airport; especially near the approach end of the active runway. I also learned that when landing on runway XX it is best to make a right 360 for spacing instead of extending your downwind for landing traffic. I don't think we would've been close to the surface area without the helicopter traffic that we had to avoid; but I will do it for safe measure next time.