The 3D contact measuring device senses work pieces in machine tools for adjusting the zero point of a numerically controlled machine tool or for positioning tasks. The device comprises a preferably elongated housing, a sensing arm projecting from the housing and bearing a sensing tip. The sensing arm being movable in the direction of a measuring axis and pivotable in all directions about the measuring axis. A sensing lever is connected to the sensing arm and forms the first universal joint, and a measuring device for detecting a parameter, which characterizes the displacement and/or the pivoting of the sensing arm. A further control surface forms a second universal joint. An elongated toothed rack of a gear transmission is aligned along the measuring axis and is movably supported and is coupled to the sensing lever via the second universal joint.