The spoon gauge is used to inspect the diameter and ____________ of the driven head once the rivet is installed.

The spoon gauge is specifically designed to assess the diameter and height of the driven head of a rivet after installation. This tool enables the mechanic to measure the rivet's head profile to ensure it meets the specifications necessary for structural integrity and proper load distribution. The height measurement is crucial as it must align with the standards set for the specific rivet type to prevent issues such as insufficient strength or failure in the assembly.

In contrast, options like width, length, and thickness do not pertain to the typical characteristics evaluated by a spoon gauge in relation to a rivet's driven head. While these dimensions might be relevant in a broader sense of material properties or other components, they do not define the specific measurements taken by the spoon gauge in this application. Hence, the focus on height in conjunction with diameter is key for ensuring that the rivet installation is within acceptable tolerances for aerospace applications.