The domes, or local elevation maxima, on the Antarctic plateau provide a unique opportunity for ground-based astronomy in that the turbulent boundary layer is so thin that a telescope on a small tower can be in the free atmosphere, i.e., the portion of the atmosphere in which the turbulence is decoupled from the effect of the Earth's surface. There, it can enjoy a free atmosphere which itself appears to offer superior conditions to that of temperate sites. This breaks the problem of characterizing the turbulence at Antarctic plateau sites into two separate tasks: determining the variability, distribution and thickness of the boundary layer, and characterizing the free atmosphere. In this article we tackle the first of these tasks using a high-resolution, low minimum sample height sonic radar (SODAR) called Snodar that has been specifically designed to characterize the Antarctic boundary thickness and structure. Snodar delivers a vertical resolution of 0.9 m, with a minimum sampling height of 8 m. Snodar sampled the first 180 m of the atmosphere with 0.9 m resolution every 10 s at Dome A, Antarctica between 2009 February 4 and 2009 August 18. The median thickness of the boundary layer over this period was 13.9 m, with the 25th and 75th percentiles at 9.7 m and 19.7 m, respectively. The data collected from Dome A also show that, while the boundary layer can be stable for several hundred hours at a time, it can also be highly variable and must be sampled on the time scale of minutes to properly characterize its thickness.