C152 Flight Instructor reported left brake failure while taxiing resulted in a taxiway excursion.

Narrative

While conducting a training flight with a student pilot; we began taxiing for departure. Immediately upon rolling; both my student and I performed brake checks. After both checks were completed; we continued the taxi with no abnormalities noted.During the taxi; we made multiple left-hand turns without issue. My student then made a right-hand turn into the run-up area; followed by an attempt to execute a left 180-degree turn to position the aircraft. At this point; he exclaimed that he had no left brake and was unable to complete the turn.I immediately took control of the aircraft and tested the brakes. The right brake functioned correctly; but the left brake had zero braking authority. As a result; I was unable to make any left turns. A right turn was not an ideal option due to the edge of the taxiway. I pulled the mixture to idle; and the aircraft rolled off into the grass straight ahead as this was the best option and there were no obstacles.The estimated groundspeed at the time was approximately 5 knots. The aircraft traveled about 5 feet into the grass before coming to a complete stop. Ground control was promptly notified of the situation. No emergency was declared; as there was no imminent threat to persons or property.Airport operations arrived with a tug; and the aircraft was safely towed to the flight school's maintenance facility. No damage occurred to the aircraft; airport property; lights; or signs. There were no injuries to any persons. A maintenance squawk was written; and I advised that the aircraft should be grounded until maintenance fixes the issue.