*knock on the door*
Enter the room quietly.
Ask, "Hey, how are you feeling?" in a genuinely concerned voice. Maybe say, "Did the pain medicine make you feel any better?"
Wait for response.
State, "The radiologist and I took a look at your 'cat' scan and it looks like you do have appendicitis."
pause momentarily for effect.
"The good news is that while this does involve surgery, it is typically an easily treatable and curative disease. I've already spoken with the surgeon and we are going to give you some antibiotics, continue to treat your pain, and admit you to the hospital for surgery tomorrow. Remember not to eat or drink anything after midnight. Do you have any questions or anything else that you need at this time?"
The key is being genuine and calm with the patient. From the beginning of the encounter all your body language and vocal tone needs to relay that everything is going to be fine and no matter what happens they are getting the best care possible. Always address pain. Smile at family members. Imagine you are an airline pilot and the patient is a passenger. They are going to "fly with you" for a few hours.
No matter what happens, as the physician, you know whats going on and how to manage it. Turbulence, appendicitis, STEMI, respiratory failure.. these are all just normal expected things that happen in the ER. If you feel this way internally your natural body language and vocal tone will convey that sentiment to the patient and there will be a calming effect.
The art of medicine.