What is the best way to calm down an elder who has agitated dementia?

Don't argue is the first thing. Then, try to distract them from the issue and redirect them to something pleasant. Use a soothing voice and calm manner. If you are upset, they will pick up on that and get more upset.

If this happens often, a trip to the doctor may be wise. Good luck. This is part of the disease and it's hard on everyone.
Carol